Caracterization of Ergodic Actions and Quantum Analogue of Noether’S Isomorphisms Theorems Souleiman Omar Hoch

Study of compact quantum groups with probabilistic methods : caracterization of ergodic actions and quantum analogue of Noether’s isomorphisms theorems Souleiman Omar Hoch

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Souleiman Omar Hoch. Study of compact quantum groups with probabilistic methods : caracterization of ergodic actions and quantum analogue of Noether’s isomorphisms theorems. Quantum Algebra [math.QA]. Université Bourgogne Franche-Comté, 2017. English. NNT : 2017UBFCD014. tel- 01905929

HAL Id: tel-01905929 https://tel.archives-ouvertes.fr/tel-01905929 Submitted on 26 Oct 2018

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Study of compact quantum groups with probabilistic methods: characterization of ergodic actions and quantum analogue of Noether’s isomorphisms theorems

submitted by Souleiman OMAR HOCH

in partial fulﬁllment of the requirements for the doctoral degrees in the Université de Bourgogne Franche-Comté - École Doctorale Carnot-Pasteur 29/06/2017 Advisor: Uwe FRANZ

Referees: Julien BICHON and Janusz WYSOCZANSKI

Examining committee:

Prof. Julien BICHON Université de Clermont-Ferrand Prof. Janusz WYSOCZANSKI University of Wroclaw Prof. Eric RICARD (PRESIDENT) Université Caen-Normandie Prof. Quanhua XU Université de Bourgogne Franche-Comté Prof. Uwe FRANZ Université de Bourgogne Franche-Comté JUNE 2017 Laboratoire de Mathématiques de Besançon École doctorale Carnot-Pasteur 16 route de Gray 25030 Besançon, France Acknowledgment

This thesis would never have been written without the contribution of many people. To all of them I wish to express my sincere gratitude. First of all, I would like to thank my advisor, Professor Uwe FRANZ, for his guidance and constant help during these fours years. I am delighted to have worked with him, because in addition to his scientiﬁc support, he has always been there to support me and advise me in the development of this thesis. It is a great honor that Professor Eric RICARD have accepted to chair my thesis and that Professor Quanhua XU agreed to be a member of my jury. It is a honor that Professor Julien BICHON, Professor Janusz WYSOCZAŃSKI have accepted to review my thesis. I would like to thank them all warmly. I also warmly thank all the members of the LMB (Besancon Mathematics Laboratory) in general and its functional analysis team especially among whom I have been trained rigorously during these last 4 years. I dedicate this thesis to my parents and my little family, Raysso, Leila and Soraya without whom my life would not have made sense. Your patience, endurance, sacriﬁces, and love keep me going. I love you all very much!

Résumé

Cette thèse étudie des problèmes liés aux treillis des sous-groupes quantiques et la carac- térisation des actions ergodiques et des états idempotents d’un groupe quantique compact. Elle consiste en 3 parties. La première partie présente des résultats préliminaires sur les groupes quantiques localement compacts, les sous-groupes quantiques normaux ainsi que les actions ergodiques et les états idempotents. La seconde partie étudie l’analogue quantique de la règle de modularité de Dedekind et de l’analogue quantique des théorèmes d’isomorphisme de Noether ainsi que leur conséquences comme le théorème de raffinement de Schreier, et le théorème Jordan-Hölder. Cette partie s’inspire du travail de recherche de Shuzhou Wang sur l’analogue quantique du troisième théorème d’isomorphisme de Noether pour les groupes quantiques compacts ainsi que le travail récent de Kasprzak, Khosravi et Soltan sur l’analogue quantique du premier théorème d’isomorphisme de Noether pour les groupes quantiques localement compacts. Dans la troisième partie, nous caractérisons les états idempotents du groupe quantique compact O−1(2) en s’appuyant sur la carac- térisation de ses actions ergodiques plongeables. Cette troisième partie est dans la ligne des travaux fait par Franz, Skalski et Tomatsu pour les groupes quantiques compacts Uq(2), SUq(2) et SOq(3). Nous classifions au préalable les actions ergodiques et les actions ergodiques plongeables du groupe quantique compact O−1(2). Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur et al. Le premier s’intitule “Fundamental isomorphism theorems for quantum groups” et a été accepté pour publication dans Expositionae Mathematicae et le second est intitulé “Ergodic actions and idempotent states of O−1(2)” et est en cours de finalisation pour être soumis.

Mots-clefs Groupe quantique localement compact, groupe quantique discret, groupe quantique linéaire- ment reductif, lemme de Zassenhaus, théorème de raﬃnement de Schreier, théorème de Jordan-Hölder, groupe quantique compact, action ergodique, état idempotent, règle de modularité de Dedekind, théorèmes d’isomorphismes de Noether.

Abstract

This thesis studies problems linked to the lattice of quantum subgroups and characterization of ergodic actions and idempotent states of a compact quantum group. It consists of three parts. The first part present some preliminary results about locally compact quantum groups, normal quantum subgroups, ergodic actions and idempotent states. The second part studies the quantum analog of Dedekind’s modularity law, Noether’s isomorphism theorem and their consequences as the Schreier refinement theorem and the Jordan-Hölder theorem. This part completes the work of Shuzhou WANG on the quantum analog of the third isomorphism theorem for compact quantum group and the recent work of Kasprzak, Khosravi and Soltan on the quantum analog of the first Noether isomorphism theorem for locally compact quantum groups. In the third part, we characterize idempotent states of the compact quantum group O−1(2) relying on the characterization of embeddable ergodic actions. This third part is in the sequence of the seminal works of Franz, Skalski and Tomatsu for the compact quantum groups Uq(2), SUq(2) and SOq(3). We classify in advance the ergodic actions and embeddable ergodic actions of the compact quantum group O−1(2). This thesis is based on two papers of the author and al. The first one is entitled “Fundamental isomorphism theorems for quantum groups” which have been accepted for publication in Expositionae Mathematicae and the second one is entitled “Ergodic actions and idempotent states of O−1(2)” and is being finalized for submission.

Keywords Locally compact quantum group, discrete quantum group, linearly reductive quantum group, Zassenhaus lemma, Schreier reﬁnement theorem, Jordan-Hölder theorem, compact quantum group, ergodic action, idempotent state, Dedekind’s modularity law, Noether’s isomorphism theorem

Contents

Acknowledgment3

Introduction 11 0.1 Brève esquisse historique...... 11 0.2 Treillis de sous-groupes quantiques...... 15 0.3 Théorèmes fondamentaux d’isomorphismes de Noether...... 18 0.4 La règle de modularité de Dedekind et lemme de Zassenhaus...... 19 0.5 Le théorème de raﬃnement de Schreier et le théorème de Jordan-Hölder pour les groupes quantiques...... 21 0.6 Caractérisation des actions ergodiques et états idempotents du groupe quantique compact O−1(2) ...... 23

Introduction 29 0.1 Brief historical outline...... 29 0.2 Lattices of quantum subgroups...... 33 0.3 Noether’s fundamental isomorphism theorems...... 35 0.4 Dedekind’s modular law and Zassenhaus lemma...... 37 0.5 The Schreier reﬁnement theorem and the Jordan-Hölder theorem for quantum groups...... 38 0.6 Characterization of ergodic actions and idempotents states of the compact quantum group O−1(2) ...... 40

1 Preliminaries 45 1.1 Preliminaries for locally compact quantum groups...... 45 1.2 Preliminaries for linearly reductive quantum groups...... 50 1.3 Lattice of closed quantum subgroups: basic facts...... 54 1.3.1 Well positioned quantum subgroups...... 66 1.4 Preliminaries on ergodic actions and idempotents states of a compact quantum group...... 68

2 Lattices of quantum subgroups of a linearly reductive quantum group 73 2.1 The second isomorphism theorem...... 73 2.2 The modular law and Zassenhaus lemma...... 76 2.3 The Schreier reﬁnement theorem...... 81 2.4 The Jordan-Hölder theorem...... 82

3 Isomorphism theorems, modular law: the locally compact case 83 3.1 The second isomorphism theorem...... 83 3.2 The third isomorphism theorem...... 86 10 Contents

3.3 The modular law and Zassenhauss lemma...... 90 3.4 Schreier and Jordan-Hölder-type results...... 94

4 Classiﬁcation results for the compact quantum group O−1(2) 97 4.1 Ergodic actions of O−1(2) ...... 97 4.2 Embeddable ergodic actions of O−1(2) ...... 100 4.3 Idempotent states of O−1(2) ...... 103 4.4 A counterexample...... 104 Introduction

Cette thèse étudie les treillis des sous-groupes normaux, la règle de modularité de Dedekind, les théorèmes d’isomorphismes de Noether et ses conséquences ainsi que la caractérisation des actions ergodiques et des états idempotents du groupe quantique compact O−1(2). La théorie des groupes quantiques commence par les travaux de Drinfeld [33], Jimbo [43][44], Reshetikhin, Takhtadzhyan et Faddeev [37] ainsi que Woronowicz [118]. Pour plus de dé- tails et d’autres aspects sur les groupes quantiques compacts, voir par exemple [15][61][57] [68]. La théorie des groupes quantiques localement compact a été introduite quant à elle dans les années 2000 par Kustermans et Vaes [66]. L’étude. Ce travail fût poursuivi par Kasprzak-Khosravi-Soltan[53]. L’autre but de cette thèse sera de prouver l’analogue quantiques des théorèmes d’isomorphismes et de leurs conséquences pour les groupes quantiques. Cette partie de la thèse s’inspire du travail de S.Natale[76] sur le théorème de Jordan-Hölder pour les algèbres de dimension ﬁnie.des états idempotents des groupes quantiques compacts et leur caractéri- sation a débuté avec les travaux de Franz-Skalski[39] et Franz-Skalski-Tomatsu[40]. Les actions ergodiques sur des algèbres d’opérateurs des groupes classiques ont été introduites dans les travaux fondateurs de Wassermann [110][108][109]. Le but de cette thèse est de poursuivre les travaux sur la caractérisation des états idempotents des groupes quantiques compacts ainsi que la caractérisation des actions ergodiques des groupes classiques. Un autre aspect d’étude des groupes quantiques compacts est de généraliser les résultats des groupes classiques aux groupes quantiques. Par exemple, Wang[107] établie l’analogue quantique du troisième théorème d’isomorphisme de Noether Cette thèse est composée de quatre chapitres. Elle repose sur les travaux[22, 23] de l’auteur et al, intitulés “Fundamental isomorphism theorems for quantum groups” et “Er- godic actions and idempotent states of O−1(2)” qui sont respectivement accepté pour publication dans Expositionae Mathematicae et en préparation. Dans cette introduction, nous rappellerons l’assise théorique de notre thèse puis nous énoncerons les résultats centraux de cette dernière.

0.1 Brève esquisse historique

Treillis de sous-groupes classiques et Théorèmes d’isomorphismes de Noether et conséquences En mathématiques, en particulier l’algèbre abstraite, les théorèmes d’isomorphisme sont trois théorèmes qui décrivent la relation entre quotients, homomorphismes et sous-objets. Des versions de ces théorèmes existent pour les groupes, les anneaux, les espaces vectoriels, les modules, les algèbres de Lie et diverses autres structures algébriques. Dans l’algèbre universelle, les théorèmes d’isomorphisme peuvent être généralisés au contexte des algèbres et des congruences. 12 Contents

Les théorèmes d’isomorphisme ont été formulés dans une certaine généralité pour les homomorphismes de modules par Emmy Noether dans son papier Abstrakter Aufbau der Idealtheorie dans algebraischen Zahl- und Funktionenkörpern qui a été publié en 1927 dans Mathematische Annalen. Des versions moins générales de ces théorèmes peuvent être trouvées dans le travail de Richard Dedekind et des articles précédents de Noether. Trois ans plus tard, B.L. Van der Waerden a publié son inﬂuent livre Algebra, le premier manuel d’algèbre abstrait qui a introduit l’aspect théorique des groupes et anneaux . Van der Waerden s’est appuyé, comme principales références pour son manuel, sur des con- férences de Noether sur la théorie des groupes et d’Emil Artin sur l’algèbre, ainsi que d’un séminaire dirigé par Artin, Wilhelm Blaschke, Otto Schreier et van der Waerden lui-même sur les idéaux. Les trois théorèmes d’isomorphisme, appelés théorème d’homomorphisme, et deux lois d’isomorphisme appliquées à des groupes, apparaissent explicitement. Cette thèse s’intéresse à l’analogue quantique des trois théorèmes d’isomorphisme dans le contexte des groupes. Notez que certaines sources commutent la numérotation des deux- ième et troisième théorèmes. Une autre variante rencontrée dans la littérature, en particulier dans l’Algèbre de Van der Waerden, est d’appeler le premier théorème d’isomorphisme le Théorème d’Homomorphisme Fondamental et par conséquent de décrémenter la numéro- tation des théorèmes d’isomorphisme restants par un. Enﬁn, dans le schéma de numérotage le plus étendu, le théorème de correspondance est parfois appelé le quatrième théorème d’isomorphisme.

L’étude des treillis de sous-groupes a une longue histoire, commençant par le travail de Richard Dedekind [31] en 1877, y compris le papier d’Ada Rottlaender [82] de 1928. Et plus tard de nombreuses contributions importantes de Reinhold Baer, Øystein Ore, Kenkichi Iwasawa, Léonid Eﬁmovitch Sadovskii, Michio Suzuki, Giovanni Zacher, Mario Curzio, Federico Menegazzo, Roland Schmidt, Stewart Stonehewer, Giorgio Busetto et beaucoup d’autres.

Zassenhaus étudia son doctorat sous la supervision d’Emil Artin. Pendant ce temps il a prouvé le lemme de Zassenhaus (papillon), un résultat magniﬁque sur des sous-groupes qui peuvent être employés pour donner une preuve simple, et très belle, du théorème de Jordan-Hölder. Il a publié ceci dans un article de 3 pages [119].

Dans la théorie des groupes, certains systèmes ordonnés de sous-groupes d’un groupe classique donné jouent un rôle important: les sous-groupes sont inclus les uns dans les autres et le système obéit à certaines conditions additionnelles. Dans cette sous-section, nous allons étudier les propriétés de ces systèmes ordonnés qu’on appellera désormais série de sous-groupes. Une série de composition fournit un moyen de décomposer une structure algébrique, comme un groupe ou un module, en morceaux simples. La nécessité de considérer les séries de compositions dans le contexte des modules résulte du fait que de nombreux modules naturels ne sont pas semi-simples et ne peuvent donc pas être décomposés en une somme directe de modules simples. Une série de composition peut ne pas exister, et quand elle le fait, elle n’a pas besoin d’être unique. Néanmoins, un groupe de résultats connus sous le nom de théorème de Jordan-Hölder aﬃrme que chaque fois que la série de composition existe, les classes d’isomorphisme de pièces simples (mais peut-être pas leur emplacement dans la série de composition en question) et leurs multiplicités sont déterminées de façon unique. 0.1. Brève esquisse historique 13

Mesure idempotente sur un groupe localement compact Soit G un groupe compact. La classe des ensembles de Borel dans G noté B est la plus petite σ-algèbre des sous-ensembles de G qui contient chaque sous-ensemble ouvert de G.

Une mesure de probabilité µ sur G est une mesure additive réelle positive vériﬁant µ(G) = 1. Si µ et ν sont deux mesures de probabilités sur G alors leur convolution µ ∗ ν est aussi une mesure de probabilité sur G. En eﬀet si X et Y sont deux variables aléatoires indépendantes sur un espace probabilisé quelconque prenant leurs valeurs dans G et si µ et ν sont leurs distributions respectives alors µ ∗ ν est la distribution du produit point par point XY . L’ensemble des mesures de probabilité sur G sera noté P(G).

Soit H un groupe compact séparé , il existe une unique mesure µH ∈ P(H) tel que:

−1 µH (E) = µH (xE) = µH (Ex) = µH (E ) pour tout ensemble de Borel E ⊂ H et x ∈ H. Cette mesure µH est appelée mesure de Haar sur H. Si H ≤ G est un sous-groupe fermé compact de G alors on appellera ωH ∈ P(G) déﬁnie par: ωH (E) = µH (E ∩ H) ∀E ∈ B. On notera désormais ωH la mesure de Haar de H.

La mesure de Haar ωG du groupe G est l’unique mesure de P(G) vériﬁant:

ωG ∗ µ = µ ∗ ωG = ωG ∀µ ∈ P(G).

Vorob’yov[104] a considéré le cas où G est un groupe commutatif fini. Hewitt et Zuckerman [41] ont étudié une classe de semi-groupes commutatifs finis incluant tous les groupes commutatifs finis. Kakehashi[50] a étudié le cas où G est le tore. Kawada et Ito[58] ont montré que les mesures idempotentes d’un groupe compact séparé découlent toutes de la mesure de Haar d’un sous-groupe fermé. Wendel[111] identifia toutes les mesures idempotentes quand G est un groupe compact séparé. Kloss[60] et Urbanik[96] ont obtenu des résultats dans le cas où G est un groupe compact séparé. Ces premières investigations ont été poursuivis par Rudin[83] et complétés par le travail de Cohen[24] qui caractérisa toutes les mesures idempotentes sur un groupe abélien localement compact.

Nous allons tout d’abord donner la déﬁnition d’une mesure idempotente avant de rappeler le résultat de Kawada-Ito.

Une mesure µ sur un groupe G est dite idempotente si µ ∗ µ = µ.

Par exemple, la mesure de Haar d’un groupe G est une mesure idempotente. Kawada et Ito ont montré que: Une mesure de probabilité sur un groupe compact séparé G est idempotente si et seulement si c’est la mesure de Haar d’un sous-groupe fermé H ≤ G.

Un groupe localement compact est un groupe topologique G tel que tout élément g ∈ G possède un voisinage compact. On supposera toujours que G possède une base dénom- brable d’ouverts.

Soit G un groupe localement compact et (X, B, µ) un espace mesuré où

µ : B → R+ ∪ {+∞} 14 Contents

est une mesure positive que nous supposerons σ-ﬁnie.

α Une action mesurable de G sur X notée (X, µ) x G est une action telle que:

(G × X → X (g, x) 7→ x est mesurable.

α L’action (X, B, µ) x G est ergodique si pour tout A ∈ B tel que gA = A pour tout g ∈ G, on a soit µ(A) = 0 soit µ(XA) = 0.

Groupes quantiques: Les débuts La notion de groupe quantique fait référence à divers objets qui sont des déformations de groupes, mais qui ont encore des propriétés très similaires aux groupes, et en particulier à des groupes de Lie semi-simples. Les plus importantes sont les algèbres de Hopf déformant les algèbres de fonction sur les groupes de Lie semi-simples ou les algèbres enveloppantes des algèbres de Lie de Kac-Moody.

Le thème populaire actuel des groupes quantiques peut être abordé à partir de deux directions essentiellement diﬀérentes. La première approche, la plus répandue, est de nature algébrique. Les premiers succès de cette approche remontent à Drinfel’d (voir [33]) et Jimbo (voir [43]), qui ont déﬁni les déformations à un paramètre des algèbres enveloppantes universelles d’algèbres de Lie complexes semisimples en 1985. Beaucoup d’autres classes de Hopf Algèbres ont été étudiées depuis 1985 et beaucoup ont reçu le label «groupe quantique». La deuxième approche est analytique dans la nature: la motivation de base dans le développement précoce de la théorie a été la généralisation de la dualité de Pontryagin pour les groupes abéliens localement compacts. Parce que le dual d’un groupe non-abélien ne peut plus être un groupe, on cherchait une catégorie plus grande contenant aussi le dual. Ces objets généralisés seraient à nouveau appelés groupes quantiques. Cette thèse traitera aussi bien de l’approche algébrique que de l’approche analytique.

La première fois que le terme « groupe quantique » est apparu à la fois de manière algébrique et de manière analytique était lors des travaux [45,46] de G.I.Kac sur les groupes d’anneaux. Kac essayait d’étendre la dualité de Pontryagin aux groupes classiques non- abéliens puisqu’en général, le dual d’un groupe classique non-abélien n’est pas forcément un groupe. Doù la nécéssité d’introduire une catégorie incluant à la fois les groupes classiques localement compacts et leurs duaux.

Dans cette optique, T.Tannaka obtint un théorème de dualité pour les groupes compacts en 1938 dans [95]. Il a réussi à recouvrir un groupe compact à partir de ses représen- tations irréductibles. Cependant ces derniers ne possedaient pas encore une structure de groupe. M.G.Krein, de son côté, déﬁnit le dual d’un groupe compact à partir de représen- tations irréductibles en les modélant sous forme d’algèbre de matrices en block.

W.F.Stinespring démontra un théorème de dualité pour les groupes localement compacts unimodulaires. Il réussit à recouvrir ce dernier à partir de l’algèbre de von Neumann de groupe en 1959. 0.2. Treillis de sous-groupes quantiques 15

En 1964, G.I.Kac&V.G.Paljutkin dans [47] donnerent le premier exemple d’un groupe quantique localement compact de dimension inﬁnie. Ils étudièrent aussi les groupes quantiques de dimension ﬁnie avec leur célèbre exemple: le groupe quantique Kac-Paljutkin de dimension 8 [48].

Indépendamment dans les années 70, G.I.Kac et L.Vainermann [49, 100] d’un coté et M.Enock et J.M.Schwartz[36] de l’autre, définirent une catégorie complète englobant les groupes localements compacts et leurs duaux. Cet objet qu’ils définirent portent de nos jours le nom d’ algèbres de Kac. S’ensuivirent plusieurs exemples d’ algèbres de Kac. Dans les années 1980, les groupes quantiques apparurent sous une forme différente par déformation quantique d’algèbres de Lie. Ces déformations sont connues sous le nom d’algèbres de Hopf. La théorie d’algèbres de Hopf est algébrique alors que l’approche des algèbres de Kac est quant-à-elle analytique.

En 1987, S.L.Woronowicz developpa la théorie des pseudo-groupes de matrices compacts[113]. Il donna comme exemple d’illustration le groupe quantique SUq(2) dans [114]. En 1998, S.L.Woronowicz déﬁni les groupes quantiques compacts dans [118]. Le point es- sentiel de sa théorie fut la démonstration de l’existence et l’unicité d’une mesure de Haar sur ces groupes quantiques compacts.

De leur part, E.Eﬀros et Z.-J.Ruan[35] et A.Van Daele [102] développerent une approche duale en déﬁnissant les groupes quantiques discrets.

En 1998, A.Van Daele [103] définit une classe de groupes quantiques, appellé groupes quantiques algébriques incluant à la fois les groupes quantiques discrets et compacts. En 2000, Johan Kustermans & Stefaan Vaes définirent les groupes quantiques localement compacts dans [66]. Il existe deux approches C∗- algébrique et Von Neumann al- gébrique d’un groupe quantique localement compact et ceux-ci généralisent les approches de l’algèbre de Kac, du groupe quantique compact et de l’algèbre de Hopf. D’autres tenta- tives d’unification de toutes ces définitions notamment avec les unitaires multiplicatives de S.Baaj et G.Skandalis [5], ont eu peu de succès à cause de leurs difficultés techniques. L’une des principales caractéristiques de cette nouvelle approche par rapport à ses prédécesseurs est l’existence axiomatique des poids invariants à gauche et à droite. Ceci donne un analogue non commutatif des mesures de Haar à gauche et à droite sur un groupe séparé localement compact. En 1996, A.Pal [78] montra que pour un groupe quantique, les états idempotents ne découlaient pas forcément de la mesure de Haar d’un sous-groupe quantique comme dans le cas classique. En 2009, U. Franz et A.Skalski[39] démontrèrent que les états idempotents d’un groupe quantique compact comoyennable sont en correspondance bijective avec les sous-algèbres coidéales avec espérance conditionnelle. Un résultat analogue pour les groupes quantiques localement compacts fut prouvé par P.Salmi et A.Skalski en 2011 dans [85].

0.2 Treillis de sous-groupes quantiques

Soit G un groupe quantique localement compact avec la comultiplication ∆G et l’unitaire ∞ multiplicatif WG. Une sous-algèbre de von Neumann N ⊂ L (G) est appelée: ∞ ¯ • coidéale à gauche si ∆G(N) ⊂ L (G) ⊗ N; ¯ • sous-algèbre invariante si ∆G(N) ⊂ N ⊗ N; 16 Contents

∞ • sous-algèbre de Baaj-Vaes si N est une sous-algèbre invariante de L (G) qui est préservée par antipode unitaire R et le groupe dilatant (τt)t∈R de G; ∗ ∞ • normale si WG(1 ⊗ N)WG ⊂ L (Gb ) ⊗¯ N; • intégrable si l’ensemble des éléments intégrables par rapport aux poids de Haar à + droite ψG est dense dans N ; En d’autres termes, la restriction de ψG à N est semiﬁnie.

∞ Dans la suite, une sous-algèbre de von Neumann de L (G) qui est une sous-algèbre coidéale à gauche sera nommée une G-coidéale ou tout simplement un coidéal. Déﬁnition 0.2.1 (Deﬁnition 1.1.9 page 49). Soit G un groupe quantique localement compact. L’ensemble de G-coidéaux sera noté CI(G). On équipe CI(G) avec la structure suivante: pour N, M ∈ CI(G), on écrit N ≤ M si N ⊂ M et N E M si N ⊂ M et N normal. L’ensemble (CI(G), ≤) admet deux opérations ∧, ∨ • N ∧ M = N ∩ M,

• N ∨ M = {xy : x ∈ N, y ∈ M}00.

(CI(G), ≤, ∧, ∨) forme un treillis qu’on appellera le treillis de coidéaux de G. Le sous-ensemble de CI(G) des G-coidéaux normaux sera noté NCI(G). Le sous- ∞ ensemble de CI(G) des sous-algèbres de Baaj-Vaes de L (G) sera noté BV(G). On peut vériﬁer que: NCI(G) and BV(G) forment des sous-treillis de CI(G). De manière similaire, NCI(G) ∩ BV(G) forme un sous-treillis de CI(G). ∞ Un groupe quantique localement compact admet un objet dual Gb = (L (Gb ), ∆ , ϕ , ψ ). Gb Gb Gb On a déﬁnit ensuite un treillis de sous-groupes quantiques comme suit:

Déﬁnition 0.2.2 (Deﬁnition 1.3.1 page 54). Soit G un groupe quantique localement compact. Le treillis BV(Gb ) sera noté QS(G) et appellé le treillis de sous-groupes quantiques de G. Le treillis NCI(Gb ) ∩ BV(Gb ) sera noté N QS(G) et appellé le treillis de sous-groupes quantiques normaux de G. On a explicité le plus grand sous-groupe quantique fermé de Gb ou en d’autres termes, la plus grande sous-algèbre de Baaj-Vaes commutative.

Proposition 0.2.3 (Proposition 1.3.3 page 55). Soit G un groupe quantique localement compact et considérons

M = {x ∈ L∞( ):(id ⊗ ∆op)(∆ (x)) = (id ⊗ ∆ )(∆ (x))}. (0.2.1) G G G G G ∞ Alors M est une sous-algèbre de Baaj-Vaes de L (G). Le groupe quantique H tel que M = ∞ L (H) est abélien. Soit N une autre sous-algèbre de Baaj-Vaes et L le groupe quantique localement compact assigné à N. Si L est abélien alors N ⊂ M. En s’appuyant sur [53, Theorem 6.2, Corollary 6.5], on a formulé le Premier Théorème d’Isomorphisme de Noether pour les groupes quantiques localement compact.

Théorème 0.2.4 (Theorem 1.3.11 page 59). Soit H et G deux groupes quantiques localement compacts, Π: → un homomorphisme et soit Π , Π , H G H/ ker Π→imΠ H/ ker Π→G Πb les homormorphismes induits par Π comme formulé sur la page 59. Alors les imdΠ→H conditionsb suivantes sont équivalentes: 0.2. Treillis de sous-groupes quantiques 17

(i) Π est un isomorphisme; H/ ker Π→imΠ ∞ ∞ ∞ (ii) L’action α :L (imΠ) → L (imΠ) ⊗¯ L (H/ ker Π) est intégrable;

(iii) ΠH/ ker Π→G identiﬁe H/ ker Π avec un sous-groupe quantique fermé de G;

(iv) Πb identifie imdΠ avec un sous-groupe quantique fermé de Hb . imdΠ→Hb Définition 0.2.5 (Definition 1.3.17 page 62). Soit G un groupe quantique localement compact et H, M ∈ QS(G). On dit que H est normalisé par M si H ∈ N QS(H ∨ M). Dans [51, Definition 2.2], Kalantar-Kasprzak-Skalski ont défini un sous-groupe quantique ouvert. Le sous-ensemble de QS(G) qui consiste des sous-groupes quantiques ouverts de G sera noté par OQS(G). On a montré en particulier que OQS(G) forme un treillis.

Proposition 0.2.6 (Proposition 1.3.26 page 64). Soit G un groupe quantique localement compact, H ∈ OQS(G) et M ∈ QS(G). Alors H ∧ M ∈ OQS(M). On a aussi prouvé une transitivité forte de l’ouverture.

Proposition 0.2.7 (Proposition 1.3.28 page 65). Soit H ≤ M ≤ G une chaîne d’inclusion fermée de groupes quantiques localement compacts. Alors, H est ouvert dans G si et seulement si H ≤ M et M ≤ G sont tous deux ouverts.

Ensuite on a déﬁni les sous-groupes quantiques bien positionnés. Pour des sous-groupes H ≤ G, nous travaillerons avec les espaces quantiques homogènes (voir Remark 1.3.2 page 54):

∞ ∞ ∞ AH = L (G/H) = cd(L (Hb )) ⊆ L (G).

Déﬁnition 0.2.8 (Deﬁnition 1.3.31 page 66). Soit H et M deux sous-groupes quantiques d’un groupe quantique localement compact G. On dit que H et M sont (relativement) bien positionnés si nous avons l’égalité

σ−cls AH ∨ AM = {AHAM} (0.2.2) En eﬀet, comme nous le verrons dans le Théorème 0.4.8, la propriété d’être bien posi- tionnés est nécessaire pour la règle de modularité des sous-groupes quantiques d’un groupe quantique localement compact. On donnera maintenant quelques conditions suﬃsantes as- surant le bien positionnement. Remarquons dans le cas algébrique, cette propriété existe toujours comme stipulé dans le Corollary 1.2.10.

Proposition 0.2.9 (Proposition 1.3.32 page 66). Les sous-groupes quantiques fermés H, M ≤ G sont relativement bien positionnés si:

(a) G est un groupe classique;

(b) l’un de H et M est compact;

(d) G est le dual d’un groupe classique. 18 Contents

0.3 Théorèmes fondamentaux d’isomorphismes de Noether

Nous allons énoncer maintenant une version quantique du Second Théorème d’Isomorphisme de Noether pour les groupes quantiques linéairement réductifs et les groupes quantiques discrets.

Théorème 0.3.1 (Theorem 2.1.4 page 75). Soit H ≤ G et K E G des sous-groupes quantiques linéairement réductifs d’un groupe quantique linéairement réductif. Si H et K génère G, alors l’homomorphisme canonique H/H ∧ K → G/K est un isomorphisme. Remarque 0.3.2. Notons qu’une version triviale du Premier Théorème d’Isomorphisme de Noether est implicite dans la preuve du Théorème 0.3.1. Pour un homomorphisme Π: H → G de groupe quantique localement compact, H/ ker Π est essentiellement le plus petit “quotient de GQLC” H →? pour lequel Π se factorise de cette manière

Π H G (0.3.1) ?

(voir par exemple Deﬁnition 1.3.8 page 58 ) D’une manière similaire, imΠ est le plus petit ? ≤ G tel que Π se factorise d’une manière similaire à (0.3.1) comme ceci

Π H G (0.3.2) ?

Dans le cas algébrique, l’image d’un homomorphisme d’algèbre de Hopf O(G) → O(H) admet clairement les deux factorisations universelles comme propriétés, et ainsi ces deux concepts coïncident par défaut. Pour cette raison, on n’a pas établit un Premier Théorème d’Isomorphisme pour les groupes quantiques linéairement réductifs ainsi que pour les groupes quantiques discrets.

On a aussi montré une version du Théorème 0.3.1 pour les groupes quantiques discrets.

Théorème 0.3.3 (Theorem 2.1.6 page 75). Si le sous-groupe H et le sous-groupe normal K génèrent le groupe quantique discret G, alors l’homomorphisme canonique H/H ∧ K → G/K est un isomorphisme. Après avoir reformulé, l’analogue quantique du Premier Théorème d’Isomorphisme de Noether (qui s’obtient trivialement dans le cas algébrique) établi dans [53, Theo- rem 6.2, Corollary 6.5] dans le contexte localement compact avec une condition supplé- mentaire d’intégrabilité, nous avons prouvé l’analogue quantique du Second Théorème d’Isomorphisme de Noether pour les groupes quantiques localement compacts toujours avec la condition supplémentaire d’intégrabilité.

Théorème 0.3.4 (Theorem 3.1.1 page 83). Soit G un groupe quantique localement compact, H ∈ QS(G) et N ∈ N QS(G). Notons Π: H → G/N l’homomorphisme induit. Alors H ∧ N = ker Π. Si de plus G = H ∨ N alors imΠ = G/N. En utilisant Théorème 0.2.4 et Théorème 0.3.4, on obtient ainsi:

Corollaire 0.3.5 (Corollary 3.1.2 page 84). L’homomorphisme H/H ∧ N → G/N est un ∞ isomorphisme si et seulement si l’action correspondante de H/H ∧ N sur L (G/N) est intégrable. 0.4. La règle de modularité de Dedekind et lemme de Zassenhaus 19

Remarque 0.3.6. Notons que les résultats dans le corollaire précédent (et donc ceux du second théorème d’isomorphisme aussi) ne sont pas vrais en général pour les groupes abéliens localement compacts classiques, comme le montre l’exemple 3.1.4. La caractéristique fondamentale de l’exemple 3.1.4 est que le produit naïf HN n’est pas fermé dans G, et ainsi H ∨ N est “plus large qu’espéré”. Même, classiquement, c’est cet échec que HN soit fermé qui empêche que les conditions du Corollaire 0.3.5 s’obtiennent. Ceci est résumé dans le résultat suivant. Proposition 0.3.7 (Proposition 3.1.5 page 84). Soit G un groupe classique localement compact et H ≤ G et N E G des sous-groupes fermés. Alors, H/H ∧ N agit de manière intégrable sur G/N si et seulement si pour tout sous-ensemble fermé F , (H ∧ N)-invariant de H le produit FN est fermé. En 2013, Shuzhou Wang démontra pour la première fois un analogue quantique de l’un des trois théorèmes d’isomorphismes de Noether, en l’occurrence l’analogue quantique du Troisième Théorème d’Isomorphisme de Noether pour les groupes quantiques compacts dans [107, Theorem 4.1]. Nous avons établi son équivalent pour les groupes quantiques localement compacts.

Théorème 0.3.8 (Theorem 3.2.6 page 89). Soit N ≤ H E G des inclusions fermées de groupes quantiques localement compacts, et supposons de plus que N est normal dans G. Alors, on a ∼ H/N E G/N and (G/N)/(H/N) = G/H. Comme le fait H/N → G/N est une inclusion fermée ne requiert pas la normalité de H, ?? 0.3.8 se généralise de la manière suivante:

Théorème 0.3.9 (Theorem 3.2.7 page 89). Soit N ≤ H ≤ G des inclusions fermées de groupes quantiques localement compacts, avec N normal dans G. Alors l’homomorphisme canonique H/N → G/N est une inclusion fermée, et

∞ ∞ L ((G/N)/(H/N)) = L (G/H)

0.4 La règle de modularité de Dedekind et lemme de Zassenhaus

Comme dans le cas classique, on a obtenu un analogue quantique du lemme de Zassenhaus pour les groupes quantiques. Cependant, il a été nécessaire tout d’abord de montrer un analogue quantique de la règle de modularité de Dedekind pour les groupes quantiques.

Proposition 0.4.1 (Proposition 2.2.1 page 76). Soit N ≤ H et M des sous-groupes normaux d’un groupe quantique linéairement réductif G. Alors, on a

H ∧ (M ∨ N) = (H ∧ M) ∨ N. (0.4.1) La règle de modularité de Dedekind s’obtient aussi pour les groupes quantiques compacts. Cependant on aura besoin de déﬁnir avant dans quel cas un sous-groupe quantique normalise un autre sous-groupe quantique. 20 Contents

Déﬁnition 0.4.2 (Deﬁnition 2.2.3 page 77). Un sous-groupe quantique L ≤ G normalise un autre M ≤ G si ce dernier est normal dans le sous-groupe quantique M ∨ L. On est maintenant prêt à énoncer.

Proposition 0.4.3 (Proposition 2.2.4 page 77). Soit G un groupe quantique compact, avec des sous-groupes quantiques L ≤ H ≤ G et M ≤ G tel que L normalise M. Alors, l’égalité H ∧ (M ∨ L) = (H ∧ M) ∨ L s’obtient. On a ensuite montré une version quantique de l’argument classique suivant: Étant donné une fonction continue f sur un groupe classique G et L ≤ G. Alors l’expression de ∞ l’espérance conditionnelle EL : L (G) → AL est donnée par: Z (ELf)(Lg) = f(lg) dl. L Lemme 0.4.4 (Lemma 2.2.5 page 78). Soit M et L deux sous-groupes quantiques d’un groupe quantique compact G tel que L normalise M. Alors:

EL(AM) ⊆ AM ∧ AL. De la même manière que pour les groupes quantiques compacts, on démontre une version quantique de la règle de modularité de Dedekind pour les groupes quantiques al- gébriques discrets en déﬁnissant au préalable un sous-groupe quantique discret normalisant un autre sous-groupe quantique discret. On a montré une version duale de la Proposition 0.4.3, pour les groupes quantiques discrets dans le sens de Deﬁnition 1.2.1. Les sous-groupes quantiques M ≤ G correspondent donc à des sous-algèbres de Hopf

kM ⊆ kG.

Déﬁnition 0.4.5 (Deﬁnition 2.2.9 page 80). Soit G un groupe quantique algébrique discret. Un sous-groupe quantique L normalise un autre M ≤ G si l’algèbre de groupe kM de ce dernier est invariante par l’action adjointe de kL sur kG. Proposition 0.4.6 (Proposition 2.2.10 page 80). Soit G un groupe quantique algébrique discret ayant les sous-groupes quantiques L ≤ H ≤ G et M ≤ G tels que L normalise M. Alors, l’égalité (0.4.1) s’obtient. Les propositions précédentes nous ont permis de démontrer une version quantique du lemme papillon (de Zassenhaus) ([62, Vol. 1, p. 77] or [81, Chapter 2, Lemma 5.10]) à la fois pour les groupes quantiques compacts et les groupes quantiques discrets.

0 0 Proposition 0.4.7 (Proposition 2.2.11 page 80). Soit A E A et B E B deux sous-groupes quantiques d’un groupe quantique soit compact soit algébriquement discret G. Alors, on a un isomorphisme 0 ∨ ( ∧ ) 0 ∨ ( ∧ ) A A B =∼ B A B . A0 ∨ (A ∧ B0) B0 ∨ (A0 ∧ B) Un résultat analogue s’obtient aussi pour les groupes quantiques linéairement réductifs G 0 dés lors que A, A , etc. sont tous normaux dans G. Pour les groupes quantiques localement compacts, l’analogue quantique de la règle de modularité de Dedekind et du lemme de Zassenhaus ne sont vrais que dans certains cas. Le théorème et la proposition suivants résument la situation. 0.5. Le théorème de raffinement de Schreier et le théorème de Jordan-Hölder pour les groupes quantiques 21

Théorème 0.4.8 (Theorem 3.3.4 page 91). Soit L ≤ H et M des sous-groupes quantiques fermés d’un groupe quantique localement compact G tel que L normalise M. Alors, on a

H ∧ (M ∨ L) = (H ∧ M) ∨ L. (0.4.2) si en plus

(a) L est compact, ou

(b) H est ouvert dans G. Remarque 0.4.9. Le lecteur doit noter que comme L = H ∧ L, la règle de modularité de Dedekind est réellement une forme de la loi de distribution H∧(M∨L) = (H∧M)∨(H∧L). Cette dernière, est néanmoins, fausse en générale.

0 0 Proposition 0.4.10 (Proposition 3.3.9 page 93). Soit A E A et B E B des sous-groupes quantiques soit

(a) compact ou

(b) ouvert d’un groupe quantique localement compact G. Alors, on a un isomorphisme 0 ∨ ( ∧ ) 0 ∨ ( ∧ ) A A B =∼ B A B . A0 ∨ (A ∧ B0) B0 ∨ (A0 ∧ B)

0.5 Le théorème de raﬃnement de Schreier et le théorème de Jordan-Hölder pour les groupes quantiques

Dans cette section, nous prouvons l’analogue quantique du théorème de raﬃnement de Schreier et le théorème de Jordan-Hölder pour les groupes quantiques compacts et discrets (voir par exemple [81, Chapter 5, Theorem 5.11] et [81, Chapter 5, Theorem 5.12]) pour l’analogue classique pour les groupes discrets ordinaires). Dans cet objectif, on a tout d’abord déﬁni l’analogue quantique de la notion de série sous-normale et d’une série de composition.

Définition 0.5.1 (Definition 2.3.1 page 81). Soit G un groupe quantique soit compact soit algébrique discret soit linéairement réductif. Un système fini

G = G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · ≥ Gk = 1 (0.5.1) de sous-groupes quantiques fermés de G est appelé une série sous-normale de G si tout sous-groupe Gi est un sous-groupe quantique normal propre de Gi−1 , i ∈ {1, 2, ··· , k}. En particulier, G1 est un sous-groupe quantique fermé normal de G, G2 est un sous-groupe quantique fermé normal de G1, mais pas nécessairement de G, et ainsi de suite. Une série sous-normale est normale si chaque Gi est normal dans le groupe ambiant G. Les groupes quantiques sous-quotients correspondants

G1\G, G2\G1, ··· , Gk\Gk−1 de G sont les facteurs de la série sous-normale (0.5.1). L’entier k est la longueur de la série (0.5.1). 22 Contents

Déﬁnition 0.5.2 (Deﬁnition 2.3.2 page 81). Une série sous-normale

G = H0 ≥ H1 ≥ H2 ≥ H3 ≥ · · · ≥ Hl = 1 (0.5.2) est appelée un raffinement de la série sous-normale (0.5.1) si chaque sous-groupe quantique Gi de (0.5.1) coïncide avec un des sous-groupes quantiques Hj, i.e. si chaque sous-groupe quantique apparaissant dans (0.5.1) apparaît aussi dans (0.5.2). En particulier, toute série normale est son propre raffinement. Les longueurs de la série normale (0.5.1) et de son raffinement (0.5.2) satisfont bien-sur l’inégalité k ≤ l. Deux séries sous-normales d’un groupe quantique compact sont dites équivalentes si leurs longueurs sont égales et leurs sous-quotients respectifs sont isomorphes à permutation près.

On est maintenant prêt à énoncer l’analogue quantique des théorèmes de raﬃnement de Schreier et de Jordan-Hölder. Leurs preuves découlent respectivement du lemme de Zassenhaus et du théorème de raﬃnement de Schreier.

Théorème 0.5.3 (Theorem 2.3.3 page 82). Deux séries sous-normales d’un groupe quantique soit compact soit discret soit linéairement réductif G admettent des raﬃnements équivalents.

Définition 0.5.4 (Definition 2.4.1 page 82). Une série sous-normale comme (0.5.1) est appelée série de composition de G si Gi est un sous-groupe quantique normal maximal propre de Gi−1 pour 1 ≤ i ≤ k. Remarque 0.5.5. En d’autres termes, une série de composition est une série sous- normale ne pouvant plus être raffinée une autre fois.

Théorème 0.5.6 (Theorem 2.4.3 page 82). Deux séries de composition d’un groupe quantique soit compact soit discret soit linéairement réductif G sont équivalents. Remarque 0.5.7. Ceci est tout simplement une adaptation au cas quantique des preuves usuelles du théorème de raﬃnement de Schreier et du théorème de Jordan-Hölder (voir par exemple les preuves de [81, Chapter 5, Theorem 5.11] et [81, Chapter 5, Theorem 5.12]). Comme mentionné ci-dessus, dès lors qu’on a obtenu une version quantique du lemme de Zassenhaus, les mêmes arguments standard marchent mécaniquement.

Pour les groupes quantiques localement compacts, l’analogue quantique du théorème de raffinement de Schreier et du théorème de Jordan-Hölder ne sont vrais aussi que dans certains cas. Notons au préalable que les Définition 0.5.1, Définition 0.5.2 et Définition 0.5.4 restent valable aussi pour les groupes quantiques localement compacts. Les deux théorèmes suivants résument la situation. On écrira {G`}`≥0 pour la série (sous)-normale générique

G = G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · ≥ Gk = 1. (0.5.3) de sous-groupes quantiques fermés d’un groupe quantique localement compact G.

Théorème 0.5.8 (Theorem 3.4.1 page 94). Soit G un groupe quantique localement com- 0 pact. Alors, deux séries sous-normales {G`} et {Gt} de G admettent des raﬃnements équivalents, dès lors que 0 G`, ` ≥ 1 et Gt, t ≥ 1 sont soit 0.6. Caractérisation des actions ergodiques et états idempotents du groupe quantique compact O−1(2) 23

(a) compacts soit

(b) ouverts.

Théorème 0.5.9 (Theorem 3.4.2 page 94). Soit G un groupe quantique localement compact. Alors, toutes les séries de composition de G consistant en des sous-groupes quantiques soit

(a) compacts soit

(b) ouverts sont équivalentes.

Les versions compacts des Théorème 0.5.8 et Théorème 0.5.9 se réfèrent aux séries sous-normales (0.5.3) dans lesquelles tous les G`, ` ≥ 1 sont compacts, mais G = G0 n’a pas besoin de l’être. Notons que ceci est équivalent au fait que le sous-quotient G/G1 est non-compact. Cependant, on a:

Proposition 0.5.10 (Proposition 3.4.3 page 95). Un groupe quantique localement compact G est compact si et seulement s’il admet une série sous-normale comme (0.5.3) avec des sous-quotients Gi/Gi+1 compacts.

0.6 Caractérisation des actions ergodiques et états idempotents du groupe quantique compact O−1(2)

Une mesure idempotente sur un groupe compact séparé classique est la mesure de Haar d’un sous-groupe fermé. Pal a montré dans [78] qu’un résultat analogue n’est en aucun cas possible dans le cas quantique en donnant l’exemple d’une mesure idempotente sur le groupe quantique compact Kac-Paljutkin de dimension 8 ne découlant pas de la mesure de Haar d’un sous-groupe quantique. Dans cette sous-section, nous allons tout d’abord définir un groupe quantique compact avant d’énoncer le résultat de Franz-Skalski [39] qui établirent en premier une caractérisation des états idempotents d’un groupe quantique compact. Ce résultat fut généralisé pour les groupes quantiques localement compact par Salmi-Skalski[85]. Dans cette section, nous allons donner la liste complète des actions ergodiques et des actions ergodiques plongeables du groupe quantique compact O−1(2) et nous caractériserons les états idempotents de ce dernier en nous appuyant sur le résultat de Franz-Skalski [39, Theorem 4.1]. Tout d’abord, notons que lister les actions ergodiques du groupe quantique compact O−1(2) revient à lister les actions ergodiques du groupe compact classique O(2). En effet, un résultat fondamental de Banica-Bichon-Collins plus précisement [6, Theorem 4.3] stipule que la catégorie de coreprésentations du groupe quantique compact O−1(n) est tenseur équivalent à la catégorie de représentation du groupe compact classique O(n). On en déduit donc que les groupes quantiques compact O−1(2) et O(2) (qui est aussi un groupe quantique compact avec q = 1) sont monoidalement équivalents. Un autre résultat fondamental de De Rijt-Vander Vennet plus précisement [30, Theorem 7.3] stipule que les actions ergodiques de deux groupes quantiques compacts monoidalement équivalents sont en correspondance bijective. On en déduit donc que les coactions ergodiques du groupe quantique compact O−1(2) sont en correspondance bijective avec les actions ergodiques du groupe compact classique O(2). La première difficulté fut donc de lister les actions 24 Contents

ergodiques du groupe compact classique O(2). Rappelons tout d’abord que si G est un α groupe compact classique, H ≤ G un sous-groupe fermé et H y N une action ergodique de H sur une algèbre de von Neumann N, on a l’action ergodique induite de G sur

G ∞ IndH (N) = {f ∈ L (G,N) | ∀g ∈ G, h ∈ H, αh(f(gh)) = f(g)} ∞ ⊆ L (G)⊗¯ N, donnée par 0 −1 0 (αg(f))(g ) = f(g g ). En particulier, si π est une représentation projective irreductible de H sur un espace de G Hilbert Vπ, on a une action ergodique de G sur une algèbre de von Neumann IndH (B(Vπ)) où B(Vπ) est équipé avec la H-action ergodique

∗ αh(x) = π(h)xπ(h) .

Par exemple, quand π est la représentation triviale (ou une représentation de dimension G ∞ une), on a IndH (C) = L (G/H) avec l’action de translation à gauche. ∼ Dans ce qui suit, on va identiﬁer O(2) = C2 n T , avec le groupe cercle T = {z ∈ C | |z| = 1}, et avec le groupe cyclique C2 = {1, σ} agissant sur T par σ(z) =z ¯. Le résultat suivant est une communication privée de Kenny De Commer.

Théorème 0.6.1 (Theorem 4.1.2 page 98). Soit O(2) M une action ergodique. Alors yα M =∼ IndO(2)(B(V )) pour un sous-groupe fermé quelconque ⊆ O(2) et π une représen- H π H tation irréductible de H. Après avoir déterminé les isomorphismes entres les actions induites, on a obtenu la liste complète des actions ergodiques non-équivalentes du groupe compact classique O(2). (∞) Les représentations projectives irréductibles de O(2) donnent soit l’action triviale β0 (∞) sur C (pour les caractères) soit l’action ergodique βl/2 = α sur M2(C) déﬁnie par ! ! ! ! a b a zlb a b d c α = , α = z c d z−lc d σ c d b a pour l ∈ N \{0} (avec l’indice pair l venant des représentations et les indices impairs l venant des représentations projectives non-triviales). Les représentations irréductibles du tore T donnent toutes la même action ergodique (∞) 2 ∞ α = α sur C = L (O(2)/T ), précisément

αz((x, y)) = (x, y), ασ(x, y) = (y, x).

Les représentations irréductibles du groupe cyclique Ck pour un certain k ﬁxé donnent (k) ∞ ∞ ∞ toutes la même action ergodique α = α on L (O(2)/Ck) = L (T/Ck) ⊕ L (T/Ck), précisément αz(f, g) = (fz, gz), ασ(f, g) = (g, f), où fz désigne le z-translaté de f. (k) Finalement, pour le groupe diédral Dk on a l’action ergodique α = β0 découlant ∞ (k) des caractères de Dk, donnant l’action sur L (O(2)/Dk), ou les actions ergodiques βl/2 0.6. Caractérisation des actions ergodiques et états idempotents du groupe quantique compact O−1(2) 25 où 0 < l < k un entier naturel découlant des représentations projectives n’étant pas des (k) caractères et βl/2 l’action ergodique induite venant de la Dk-représentation projective ! ! ! ! a b a zlb a b d c α = , α = . z c d z−lc d σ c d b a On a obtenu ainsi: Proposition 0.6.2 (Proposition 4.1.3 page 100). La liste complète des actions ergodiques non-équivalentes de O(2):

(k) (∞) (∞) (k0) 0 0 {βl/2 , βl0/2 , α , α | k, k ∈ N0, l ∈ N, 0 ≤ l < k}.

On veut en déduire la liste des actions ergodiques plongeables de O−1(2). Pour cela, nous procédons comme suit. Notons tout d’abord que par définition, une action ergodique plongeable est une ∗- algèbre de comodule de l’algèbre de groupe quantique compact A−1 associée à O−1(2) qui s’injecte dans A−1 comme telle (c-à-d par une inclusion qui préserve toutes les structures d’algèbre, de comodule, etc). Puisque l’équivalence twistant λ. (voir la notation de Section 4.4) qui implémente The- orem 1.4.19 implémente aussi une équivalence entre les catégories de ∗-algèbres coidéales sur A−1 et la version non-twistée A (algèbre des fonctions représentatives sur le groupe classique O(2)), il sera suffisant d’identifier l’O(2)-action ergodique B dans la liste de Proposition 0.6.2 pour laquelle ∼ λ . B = A−1 comme ∗-algèbres de comodule A−1, et d’identifier aussi alors les membres de cette liste s’injectant dans B. (2) On a montré qu’il n’y a qu’un candidat pour B (précisement β1/2) en utilisant le théorème de Peter-Weyl pour déterminer le type de représentation des actions ergodiques identifiées dans la Proposition 0.6.2 (où par représentation type, on sous-entend les mul- tiplicités des différentes O(2)-représentations irréductibles). Ceci est en l’occurrence la substance de notre résultat suivant: Proposition 0.6.3 (Proposition 4.2.1 page 100). Les seules algèbres de comodule parmi celles listées dans la Proposition 0.6.2 qui sont isomorphiques à A comme O(2)-représentations (1) ∼ (2) sont α = A elle-même et β1/2. On a ainsi obtenu comme conséquence la liste complète des actions ergodiques plongeables du groupe quantique compact O−1(2): Corollaire 0.6.4 (Corollary 4.2.2 page 101). L’équivalence twistant λ. induit une bijection entre (k) (k) {α , βl/2 | k = ∞ ou pair , l = 0 ou impair } de Proposition 0.6.2 et l’isomorphisme de classe des actions ergodiques plongeables de O−1(2).

La liste complète des coactions ergodiques plongeables non-équivalentes de O−1(2) est donc: (k) (k) {α , βl/2 | k = ∞ ou pair , l = 0 ou impair }

Pour ﬁnir cette section, on s’est posé la question suivante motivée par Theorem 1.4.19 page 71: 26 Contents

Question 0.6.5 (Question 1.4.1 page 71). Soit G1 et G2 deux groupes quantiques compacts monoidalement équivalents. A-t-on une correspondance bijective entre leurs actions ergodiques plongeables? Remarque 0.6.6. Une réponse aﬃrmative à la question précédente élargira le résultat fondamental [30, Theorem 7.3] de De Rijt-Vander Vennet aux actions ergodiques plongeables. Mais la réponse est négative en général (Voir Question 0.6.7 ci-dessous) sauf dans le cas où nous avons une condition supplémentaire ( Voir Proposition 4.4.5 page 106).

Spéciﬁquement, supposons que G1 et G2 sont deux groupes quantiques compacts monoidalement équivalents. Supposons de plus que l’équivalence monoidale est implé- menté par un twist avec un cocycle

⊗2 λ : C(G1) → C. Alors, d’après la preuve de Theorem 1.4.19, on a une opération de twist λ. qui implémente une équivalence entre la catégorie Erg(G1) des G1-actions ergodiques et la catégorie analogue Erg(G2). Dans cet esprit, la formulation de la Question 0.6.5 à laquelle nous avons répondu est: Question 0.6.7 (Question 4.4.1 page 104). Peut-on restreindre

λ. : Erg(G1) → Erg(G2) à une équivalence entre les sous-catégories des coactions ergodiques plongeables?

Malheureusement, la réponse à la Question 0.6.7 est négative en générale, comme en l’a montré dans le contre-exemple ci-dessous. Prenons le groupe quantique compact G1 comme étant le groupe de Heisenberg d’ordre 64 4 4 4 H4 := hε1, ε2, δ | ε1 = ε2 = δ = 1, δε1 = ε1δ, ε2δ = δε2, ε1ε2 = δε2ε1i.

Sa déformation G2 est celle utilisée dans [16] et décrite en détail dans la section 6 de [16]. Cependant, on n’écrira pas explicitement ici le cocycle

⊗ λ : C(H4) → C, rappelons juste qu’il est obtenu comme la composition

λ ⊗2 C(H4) C C(Γ)⊗2 ou la ﬂèche du membre de gauche est la restriction de l’inclusion 2 ∼ 2 2 (Z/2) = Γ := hε1, ε2i ⊂ H4 et la ﬂèche du membre de droite est un certain cocycle induisant la classe de cohomologie non-nulle dans 2 × ∼ H (Γb, C ) = Z/2.

Il est plus habile de dualiser et de travailler avec l’algèbre de groupe CH4 et sa version dé- ∗ formée H , qui coïncide avec CH4 comme algèbre mais ayant une comultiplication twistée. On a montré que: 0.6. Caractérisation des actions ergodiques et états idempotents du groupe quantique compact O−1(2) 27

Proposition 0.6.8 (Proposition 4.4.3 page 106). Il n’existe pas de bijection préservant l’inclusion, la dimension, et l’action de G entre les sous-algèbres coidéales de CH4 et ceux H∗.

Comme dernière observation, on a montré que s’il arrive que G1 admet un ensemble 2 1 de 2-cohomologie H (Gc1, S ) trivial, la réponse à la Question 0.6.7 est aﬃrmative:

Proposition 0.6.9 (Proposition 4.4.5 page 106). Soit G1 et G2 deux groupes quantiques 2 1 compacts monoidalement équivalents, et supposons que H (Gc2, S ) est trivial. Alors,

λ. : Erg(G1) → Erg(G2) se restreint à une équivalence entre les sous-catégories des coactions ergodiques plongeables.

États idempotents du groupe quantique compact O−1(2)

Ayant réussi à faire la liste des actions ergodiques plongeables de O−1(2), vu que par déf- inition, une action ergodique plongeable s’identiﬁe à une sous-algèbre coidéale et vu que d’après le résultat fondamental [39, Theorem 4.1] de Franz-Skalski, il y a une correspondance bijective entre les sous-algèbres coidéaux avec espérance conditionnelle et les états idempotents d’un groupe quantique compact; notre projet s’est heurté à un dernier souci: Comment exprimer par une formule concrète un état idempotent à partir d’une action d’un groupe quantique compact? En fait, bien que nous les ayons caractérisées grâce à [39], il nous reste à lister les états idempotents de O−1(2) à partir de la liste de ses actions ergodiques plongeables. Vu que:

1. Rappelons que, si on prend la comultiplication elle-même comme l’action (c-à-d on prend le groupe quantique lui-même comme la sous-algèbre coidéale), ce n’est pas immédiat de produire le correspondent état de Haar depuis la comultiplication.

2. Aussi si on considère la sous-algèbre C, alors on voit qu’on ne peut pas produire l’état de Haar depuis l’action qui est simplement C → C ⊗ C, 1 7→ 1 ⊗ 1. Le problème est que partant de la preuve de [39, Theorem 4.1], il y a plusieurs éléments que nous avons abstraitement mais de manière non-explicite, et ils dépendent les uns des autres. En eﬀet, pour obtenir l’état idempotent de la sous-algèbre coidéale, on a besoin de l’espérance conditionnelle E; Et pour obtenir E, on a malheureusement besoin de l’état dans la preuve de [39, Theorem 4.1]. C’est donc circulaire et ne conduit pas à quelque chose d’ explicite (juste au théorème de l’article Franz-Skalski, c-à-d le fait qu’il y a cette bijection). Nous pensons que s’il y a une chance d’obtenir une formule concrète de l’état idempotent à partir d’une action, ce serait en exprimant l’espérance conditionnelle E seulement en fonction de l’action ergodique plongeable α en ne faisant pas intervenir l’état lui-même, puis de composer E ainsi obtenue avec la counité ε.

La question suivante résume notre point de vue:

Question 0.6.10 (Question 4.3.1 page 104). Est-il possible d’obtenir une telle formule concrète en généralité pour un groupe quantique compact au moins dans le cas Kac?

Introduction

This thesis studies the lattice of subgroups, Dedekind’s modular law, Noether’s isomorphism theorems and its consequences, as well as the characterization of the ergodic actions and idempotent states of the compact quantum group O−1(2). The theory of quantum groups begins with the work of Drinfeld [33], Jimbo [43][44], Reshetikhin, Takhtadzhyan and Faddeev [37] and Woronowicz [118]. For more details and other aspects on compact quantum groups, see for example [15][61][57][68]. The theory of locally compact quantum groups was introduced in the 2000s by Kustermans and Vaes [66]. The study of the idempotent states of the compact quantum groups and their characterization began with the work of Franz-Skalski [39] and Franz-Skalski-Tomatsu [40]. The ergodic actions on algebras of operators of classical groups have been introduced in the founding works of Wassermann [110][108][109]. The aim of this thesis is to continue the work on the characterization of the idempotent states of the compact quantum groups as well as the characterization of the ergodic actions of the classical groups. Another aspect of the study of compact quantum groups is to generalize the results of the classical groups to the quantum groups. For example, Wang [107] establishes the quantum analogue of Noether’s third isomorphism theorem. This work was pursued by Kasprzak-Khosravi-Soltan [53]. The other goal of this thesis will be to prove the quantum analogue of isomorphism theorems and their consequences for quantum groups. This part of the thesis is inspired by the work of S.Natale [76] on the theorem of Jordan-Hölder for ﬁnite-dimensional algebras. This thesis is composed of four chapters. It is based on the author’s [22, 23] works and al, entitled “Fundamental isomorphism theorems for quantum groups ” and “Ergodic actions and idempotent states of O1(2) ” which are respectively accepted for publication in Expositionae Mathematicae and in preparation. In this introduction, we will recall the theoretical basis of our thesis and then we will state the central results of the latter.

0.1 Brief historical outline

Lattices of classical subgroups and Noether’s Isomorphisms Theorems and consequences. In mathematics, in particular abstract algebra, isomorphism theorems are three theorems which describe the relation between quotients, homomorphisms and subobjects. Versions of these theorems exist for groups, rings, vector spaces, modules, Lie algebras and various other algebraic structures. In universal algebra, the isomorphic theorems can be generalized to the context of algebras and congruences. The isomorphims theorems have been formulated in a certain generality for the homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie 30 Contents

in algebraischen Zahl- und Funktionenkörpern which was published in 1927 in Mathematis- che Annalen. Less general versions of these theorems can be found in the work of Richard Dedekind and previous papers by Noether. Three years later, B. Van der Waerden published his influential book Algebra, the first abstract algebra manual that introduced the theoretical aspect of groups and rings. Van der Waerden relied, as the main references for his textbook, on Noether’s lectures on group theory and Emil Artin lectures on algebra, as well as a seminar led by Artin, Wilhelm Blaschke, Otto Schreier and Van der Waerden himself on ideals. The three isomorphism theorems, called the homomorphism theorem, and two isomorphism laws applied to groups, appear explicitly. This thesis deals with the quantum analogue of the three isomorphism theorems in the context of groups. Note that some sources switch the numbering of the second and third theorems. Another variant encountered in the literature, particularly in Van der Waer- den’s Algebra, is to call the first isomorphism theorem the Theorem of Fundamental Ho- momorphism and consequently to decrement the numbering of the remaining isomorphism theorems by one. Finally, in the most extended numbering scheme, the correspondence theorem is sometimes called the fourth isomorphism theorem.

The study of subgroup lattices has a long history, beginning with the work of Richard Dedekind [31] in 1877, including the 1928 Ada Rottlaender paper [82]. And later many important contributions of Reinhold Baer, Øystein Ore, Kenkichi Iwasawa, Leonid Eﬁ- movich Sadovskii, Michio Suzuki, Giovanni Zacher, Mario Curzio, Federico Menegazzo, Roland Schmidt, Stewart Stonehewer, Giorgio Busetto and many others.

Zassenhaus studied his thesis under the supervision of Emil Artin. Meanwhile he proved the Zassenhaus lemma (butterﬂy), a magniﬁcent result on subgroups that can be used to give a simple, and very beautiful proof of the Jordan-Hölder theorem. He has published this in a 3 pages paper [119].

In group theory, some ordered systems of subgroups of a given classical group play an important role: subgroups are included in one another and the system obeys certain additional conditions. In this subsection, we will study the properties of these ordered systems, which will now be called series of subgroups. A composition serie provides a means of decomposing an algebraic structure, such as a group or a module, into simple pieces. The need to consider the series of compositions in the context of modules results from the fact that many natural modules are not semi-simple and therefore can not be decomposed into a direct sum of simple modules. A composition serie may not exist, and when it does, it does not need to be unique. Nevertheless, a group of results known as the Jordan-Hölder theorem asserts that whenever the composition series exists, the isomorphism classes of simple pieces (but perhaps not their location in the composition series) and their multiplicities are determined in a unique way.

Idempotent measure on a locally compact group Let G be a compact group. The class of Borel sets in G denoted by B is the smallest σ-algebra of the subsets of G which contains each open subset of G.

A probability measure µ on G is a positive real additive measure satisfying µ(G) = 1. If µ and ν are two probability measures on G then their convolution µ∗ν is also a probability measure on G. 0.1. Brief historical outline 31

In fact if X and Y are two independent random variables on any probability space taking their values in G and if µ and ν are their respective distributions then µ ∗ ν is the distribution of the point by point product XY . The set of probability measures on G will be denoted P(G).

Let H be a Hausdorﬀ compact group, there exists a unique measure µH ∈ P(H) such that: −1 µH (E) = µH (xE) = µH (Ex) = µH (E ) for all Borel set E ⊂ H and x ∈ H. This measure µH is called Haar measure on H. If H ≤ G is a compact closed subgroup of G then ωH ∈ P(G) is deﬁned by: ωH (E) = µG(E ∩ H) ∀E ∈ B. We will henceforth note ωH the Haar measure of H.

The Haar measure ωG of the group G is the unique measure of P(G) verifying:

ωG ∗ µ = µ ∗ ωG = ωG ∀µ ∈ P(G).

Vorob’yov [104] considered the case where G is a finite commutative group. Hewitt and Zuckerman [41] studied a class of finite commutative semigroups including all finite commutative groups. Kakehashi [50] studied the case where G is the circle group. Kawada and Ito [58] have shown that all idempotent measures of a Hausdorff compact group arises from the Haar measure of a closed subgroup. Wendel [111] identified all the idempotent measures when G is a separable compact group. Kloss [60] and Urbanik [96] have obtained some results in the case where G is a Hausdorff compact group. These first investigations were continued by Rudin [83] and completed by the work of Cohen [24] who characterized all idempotent measures on a locally compact abelian group.

We will ﬁrst deﬁne an idempotent measure before recalling Kawada-Ito result.

A measure µ on a group G is called idempotent if µ ∗ µ = µ.

For example, the Haar measure of a group G is an idempotent measure. Kawada and Ito proved that: A probability measure on a Hausdorﬀ compact group G is idempotent if and only if it is the Haar measure of a closed subgroup H ≤ G.

A locally compact group is a topological group G such that every element g ∈ G has a compact neighborhood. We always assume that G has a countable base of open sets.

Let G be a locally compact group and (X, B, µ) a measured space where

µ : B → R+ ∪ {+∞} is a positive measure that we assume σ-ﬁnite.

α A measurable action of G on X denoted by (X, µ) x G is an action such that: (G × X → X (g, x) 7→ x is µ-measurable.

α The action (X, B, µ) x G is ergodic if for any A ∈ B such that gA = A for all g ∈ G, we have either µ(A) = 0 or µ(X \ A) = 0. 32 Contents

Quantum groups: The early days The notion of quantum group refers to various objects which are deformations of groups but which still have very similar properties to groups, and in particular to semi-simple Lie groups. The most important are the Hopf algebras which deform function algebras on semi-simple Lie groups or enveloping algebras of Kac-Moody Lie algebras.

The current popular theme of quantum groups can be approached from two essentially different directions. The first approach, the most widespread, is algebraic in nature. The first successes of this approach date back to Drinfel’d (see [33]) and Jimbo (see [43]), who defined the deformations with one parameter of a universal enveloping algebra of semisimple complex Lie algebras in 1985. Many other classes of Hopf Algebras have been studied since 1985 and many have received the label "quantum group". The second approach is analytical in nature: the basic motivation in the early development of the theory has been the generalization of Pontryagin duality for locally compact abelian groups. Because the dual of a non-abelian group can no longer be a group, we are looking for a larger category that also contains the dual. These generalized objects would again be called quantum groups. This thesis will deal with both the algebraic approach and the analytical approach.

The ﬁrst time that the term "quantum group" appeared both algebraically and analyt- ically was during G.I.Kac [45,46] work on the ring groups. Kac tried to extend the duality of Pontryagin to non-abelian classical groups, since in general the dual of a classical non- abelian group is not necessarily a group. Hence the need to introduce a category including both locally compact classical groups and their duals.

From this perspective, T. Tannaka obtained a duality theorem for compact groups in 1938 in [95]. He managed to recover a compact group from its irreducible representations. However, the latter did not yet have a group structure. M.G. Krein, for his part, deﬁned the dual of a compact group from irreducible representations by modeling them in the form of an algebra of block matrix.

W.F.Stinespring proved a duality theorem for locally compact unimodular groups. He succeeded to recover the latter from the group von Neumann algebra in 1959.

In 1964, G.I.Kac & V.G.Paljutkin in [47] gave the first example of a locally compact quantum group of infinite dimension. They also studied quantum groups of finite dimension with their famous example: the Kac-Paljutkin quantum group of dimension 8 [48].

Independently in the 1970s, G.I.Kac and L.Vainermann [49,100] on the one hand and M.Enock and J.M.Schwartz [36] on the other, defined a complete category encompassing locally compact groups and their duals. This object, which they define, now bears the name of Kac algebras. Several examples of Kac algebras followed. In the 1980s, quantum groups appeared in a different form by quantum deformation of Lie algebras. These deformations are known as Hopf algebras. The theory of Hopf algebras is algebraic whereas the approach of Kac algebras is analytical.

In 1987, S.L.Woronowicz developed the theory of compact matrix pseudo-groups [113]. He gave as an illustration example the quantum group SUq(2) in [114]. In 1998, S.L.Woronowicz deﬁned the compact quantum groups in [118]. An essential point of his theory was the 0.2. Lattices of quantum subgroups 33

demonstration of the existence and the uniqueness of a Haar measure on these compact quantum groups.

On the other hand, E.Eﬀros and Z.-J.Ruan [35] and A.Van Daele [102] developed a dual approach by deﬁning the discrete quantum groups.

In 1998, A.Van Daele [103] defines a class of quantum groups, called algebraic quantum groups including both discrete and compact quantum groups. In 2000, Johan Kustermans & Stefaan Vaes defined locally compact quantum groups in [66]. There is two approaches to locally compact quantum groups: C∗-algebraic approache and Von Neumann algebraic approache, and these generalize the approaches to the Kac algebras, the compact quantum group, and the Hopf algebras. Other attempts to unify all these definitions, particularly with the multiplicatives unitaries of S.Baaj and G.Skandalis [5], have had little success because of their technical difficulties. One of the main characteristics of this new approach compared to its predecessors is the axiomatic existence of invariant weights on the left and right. This gives a non-commutative analogue of the left and right Haar measure on a Hausdorff locally compact group. In 1996, A.Pal [78] showed that for a quantum group the idempotent states did not necessarily result from the Haar measure of a quantum subgroup as in the classical case. In 2009, U. Franz and A.Skalski [39] proved that the idempotent states of a compact quantum group are in bijective correspondence with the coideal subalgebras with conditional expectation. A similar result for locally compact quantum groups was proved by P.Salmi and A.Skalski in 2011 in [85].

0.2 Lattices of quantum subgroups

Let G be a locally compact quantum group with comultiplication ∆G and multiplicative ∞ unitary WG. A von Neumann subalgebra N ⊂ L (G) is called ∞ ¯ • Left coideal if ∆G(N) ⊂ L (G) ⊗ N; ¯ • Invariant subalgebra if ∆G(N) ⊂ N ⊗ N; ∞ • Baaj-Vaes subalgebra if N is an invariant subalgebra of L (G) which is preserved by unitary antipode R and the scaling group (τt)t∈R of G; ∗ ∞ • Normal if WG(1 ⊗ N)WG ⊂ L (Gb ) ⊗¯ N;

Deﬁnition 0.2.1 (Deﬁnition 1.3.1 page 54). Let G be a locally compact quantum group. The set of G-coideals will be denoted by CI(G). We equip CI(G) with the poset structure: for N, M ∈ CI(G) we write N ≤ M if N ⊂ M. The poset (CI(G), ≤) admits two operations ∧, ∨

• N ∧ M = N ∩ M,

• N ∨ M = {xy : x ∈ N, y ∈ M}00. 34 Contents

(CI(G), ≤, ∧, ∨) forms a lattice which will be called the lattice of coideals of G. The subset of CI(G) of normal G-coideals will be denoted NCI(G). The subset of ∞ CI(G) of Baaj-Vaes subalgebras of L (G) will be denoted BV(G). It is easy to check that NCI(G) and BV(G) form sublattices of CI(G). Similarly NCI(G) ∩ BV(G) forms a sublattice of CI(G).

∞ A locally compact quantum group admits a dual object Gb = (L (Gb ), ∆ , ϕ , ψ ). Gb Gb Gb A lattice of quantum subgroups was then deﬁned as follows:

Deﬁnition 0.2.2. Let G be a locally compact quantum group. The lattice BV(Gb ) will be denoted QS(G) and called a lattice of quantum subgroups of G. The lattice NCI(Gb ) ∩ BV(Gb ) will be denoted N QS(G) and called a lattice of normal quantum subgroups of G.

We make explicit the largest closed quantum subgroup of Gb or in other terms, the largest commutative Baaj-Vaes subalgebra.

Proposition 0.2.3 (Proposition 1.3.3 page 55). Let G be a locally compact quantum group and let us consider

M = {x ∈ L∞( ):(id ⊗ ∆op)(∆ (x)) = (id ⊗ ∆ )(∆ (x))}. (0.2.1) G G G G G

∞ ∞ Then M is a Baaj-Vaes subalgebra of L (G). The quantum group H such that M = L (H) is abelian. Let N be another Baaj-Vaes subalgebra and L be the locally compact quantum group assigned to N. If L is abelian then N ⊂ M. Relying on [53, Theorem 6.2, Corollary 6.5], we formulate the First Noether’s Isomor- phism Theorem for locally compact quantum group.

Theorem 0.2.4 (Theorem 1.3.11 page 59). Let H and G be locally compact quantum groups, Π: H → G a morphism and let Π / ker Π→imΠ, ΠH/ ker Π→G, Πb be the mor- H imdΠ→H phisms induced by Π as described above. Then the following conditions are equivalent:b

(i) Π is an isomorphism; H/ ker Π→imΠ

∞ ∞ ∞ (ii) the action α :L (imΠ) → L (imΠ) ⊗¯ L (H/ ker Π) is integrable;

(iii) ΠH/ ker Π→G identiﬁes H/ ker Π with a closed quantum subgroup of G;

(iv) Πb identifies imdΠ with a closed quantum subgroup of Hb . imdΠ→Hb Definition 0.2.5 (Definition 1.3.17 page 62). Let G be a locally compact quantum group and H, M ∈ QS(G). We say that H is normalized by M if H ∈ N QS(H ∨ M). In [51, Definition 2.2], Kalantar-Kasprzak-Skalski defined an open quantum subgroup. The subset of QS(G) that consists of open quantum subgroups of G will be denoted by OQS(G). We showed in particular that OQS(G) forms a lattice.

Proposition 0.2.6 (Proposition 1.3.26 page 64). Let G be a locally compact quantum group, H ∈ OQS(G) and M ∈ QS(G). Then H ∧ M ∈ OQS(M). We proved also a strong transitivity of openness. 0.3. Noether’s fundamental isomorphism theorems 35

Proposition 0.2.7 (Proposition 1.3.28 page 65). Let H ≤ M ≤ G be a chain of closed embeddings of locally compact quantum groups. Then, H is open in G if and only if

H ≤ M and M ≤ G are both open.

After that, we deﬁne the well-positioned quantum subgroup. For subgroups H ≤ G, we will be working with the quantum homogeneous spaces (see Remark 1.3.2 page 54)

∞ ∞ ∞ AH = L (G/H) = cd(L (Hb )) ⊆ L (G).

Deﬁnition 0.2.8. Let H and M be two closed quantum subgroups of a locally compact quantum group G. We say that H and M are (relatively) well positioned if we have the equality σ−cls AH ∨ AM = {AHAM} (0.2.2) (or equivalently its analogue with H and M reversed). As we will see in Theorem 3.3.4, the well positioning property is relevant to the modular law for quantum subgroups of a locally compact quantum group. Here, we discuss suﬃcient conditions that ensure well positioning. Let us also note that in the algebraic context the counterpart of well positioning always holds as noted in Corollary 1.2.10.

Proposition 0.2.9 (Proposition 1.3.32 page 66). The closed quantum subgroups H, M ≤ G are relatively well positioned if

(a) G is classical;

(b) one of H and M is compact;

(d) G is dual-classical.

0.3 Noether’s fundamental isomorphism theorems

We shall now state a quantum version of Noether’s Second Isomorphism Theorem for linearly reductive quantum groups and discrete quantum groups.

Theorem 0.3.1 (Theorem 2.1.4 page 75). Let H ≤ G and K E G be linearly reductive quantum subgroups of a linearly reductive quantum group. If H and K generate G, then the canonical morphism H/H ∧ K → G/K is an isomorphism. Remark 0.3.2. Note incidentally that a trivial version of the ﬁrst isomorphism theorem is implicit in the proof of Theorem 2.1.4. For a morphism Π: H → G of locally compact quantum groups, H/ ker Π is essentially the smallest “quotient LCQG” H →? for which Π factors as Π H G (0.3.1) ? 36 Contents

(see e.g. Deﬁnition 1.3.8 page 58) Similarly, imΠ is the smallest ? ≤ G such that Π factors similarly to (2.1.3) as

Π H G (0.3.2) ?

In the algebraic case, the image of a Hopf algebra morphism O(G) → O(H) clearly has both factorization universality properties, and hence by default the two concepts coincide. For this reason, we do not state a First Isomorphism Theorem in the present section.

We have also shown a version of the Theorem 0.3.1 for discrete quantum groups.

Theorem 0.3.3 (Theorem 2.1.6 page 75). If the subgroup H and the normal subgroup K generate the discrete quantum group G, then the canonical morphism H/H ∧ K → G/K is an isomorphism.

After we reformulated the quantum analogue of the Noether First Isomorphism The- orem (which is obtained trivially in the algebraic case) established in [53, Theorem 6.2, Corollary 6.5] in the locally compact context with an additional condition of integrability, we proved the quantum analogue of the Noether’s Second Isomorphism Theorem for locally compact quantum groups always with the additional condition of integrability.

Theorem 0.3.4 (Theorem 3.1.1 page 83). Let G be a locally compact quantum group, H ∈ QS(G) and N ∈ N QS(G). Let us denote by Π: H → G/N the induced morphism. Then H ∧ N = ker Π. If moreover G = H ∨ N then imΠ = G/N. Using Theorem 1.3.11 and Proposition 3.1.1 we get

Corollary 0.3.5 (Corollary 3.1.2 page 84). The homomorphism H/H ∧ N → G/N is an ∞ isomorphism if and only if the corresponding action of H/H∧N on L (G/N) is integrable. Remark 0.3.6. Let us also note that equivalent statements in Corollary 0.3.5 fails (and hence so does the second isomorphism theorem) in general even classically, for locally compact abelian groups, as the Example 3.1.4 page 84 shows. The fundamental characteristic of Example 3.1.4 page 84 is that the naive product HN is not closed in G, and hence H ∨ N is “larger than expected”. Indeed, classically, it is this failure of HN to be closed that prevents the conditions of Corollary 0.3.5 from holding. This is summarized in the following result.

Proposition 0.3.7 (Proposition 3.1.5 page 84). Let G be a classical locally compact group, and H ≤ G and N E G closed subgroups. Then, H/H∧N acts integrably on G/N if and only if for every (H∧N)-invariant closed subset F of H the product FN is closed. In 2013, Shuzhou Wang demonstrated for the ﬁrst time a quantum analogue of one of Noether’s three isomorphism theorems, namely the quantum analogue of the Noether Third Isomorphism Theorem for compact quantum groups in [107, Theorem 4.1]. We have established its equivalent for locally compact quantum groups.

Theorem 0.3.8 (Theorem 3.2.6 page 89). Let N ≤ H E G be inclusions of closed locally compact quantum subgroups, and assume furthermore that N is normal in G. Then, we have ∼ H/N E G/N and (G/N)/(H/N) = G/H. 0.4. Dedekind’s modular law and Zassenhaus lemma 37

Since the conclusion that H/N → G/N is a closed embedding does not actually require the normality of H, Theorem 0.3.8 is generalized in the following way:

Theorem 0.3.9 (Theorem 3.2.7 page 89). Let N ≤ H ≤ G be closed embeddings of locally compact quantum groups, with N normal in G. Then the canonical morphism H/N → G/N is a closed embedding, and ∞ ∞ L ((G/N)/(H/N)) = L (G/H)

0.4 Dedekind’s modular law and Zassenhaus lemma

As in the classical case, a quantum analogue of the Zassenhaus lemma has been obtained for quantum groups. However, it was ﬁrst necessary to show a quantum analogue of Dedekind’s modular law for quantum groups.

Proposition 0.4.1 (Proposition 2.2.1 page 76). Let N ≤ H and M be normal subgroups of the linearly reductive quantum group G. Then, we have H ∧ (M ∨ N) = (H ∧ M) ∨ N. (0.4.1) Dedekind’s modular law is also obtained for compact quantum groups. However, it was necessary to deﬁne before when a quantum subgroup normalizes another quantum subgroup.

Deﬁnition 0.4.2 (Deﬁnition 2.2.3 page 77). A quantum subgroup L ≤ G normalizes another M ≤ G if the latter is normal in the quantum subgroup M ∨ L. We are now ready to state

Proposition 0.4.3 (Proposition 2.2.4 page 77). Let G be a compact quantum group, with quantum subgroups L ≤ H ≤ G and M ≤ G such that L normalizes M. Then, the equality H ∧ (M ∨ L) = (H ∧ M) ∨ L holds. After this, we prove a quantum version of the following classical argument: ∞ Given a continuous function f on G, the expression for its expectation EL : L (G) → A is L Z (ELf)(Lg) = f(lg) dl. L Lemma 0.4.4 (Lemma 2.2.5 page 78). Let M and L be quantum subgroups of a compact quantum group G such that L normalizes M. Then, we have

EL(AM) ⊆ AM ∧ AL. In the same way as for compact quantum groups, we demonstrate a quantum version of Dedekind’s modular law for discrete algebraic quantum groups by deﬁning a discrete quantum subgroup normalizing another discrete quantum subgroup. We have shown a dual version of the previous Proposition 0.4.3, for discrete quantum groups in the sense of Deﬁnition 1.2.1. Quantum subgroups M ≤ G then correspond to Hopf subalgebras kM ⊆ kG. 38 Contents

Definition 0.4.5 (Definition 2.2.9 page 80). Let G be an algebraic discrete quantum group. A quantum subgroup L normalizes another M ≤ G if the group algebra kM of the latter is invariant under the adjoint action of kL on kG. Proposition 0.4.6 (Proposition 2.2.10 page 80). Let G be an algebraic discrete quantum group, with quantum subgroups L ≤ H ≤ G and M ≤ G such that L normalizes M. Then, the equality (2.2.1) holds. The previous propositions will allow us to prove the following version of the butterfly (or Zassenhaus) lemma ([62, Vol. 1, p. 77] or [81, Chapter 5, Lemma 5.10]) for compact and discrete quantum groups.

0 0 Proposition 0.4.7 (Proposition 2.2.11 page 80). Let A E A and B E B be quantum subgroups of either a compact or an algebraic discrete quantum group G. Then, we have an isomorphism 0 ∨ ( ∧ ) 0 ∨ ( ∧ ) A A B =∼ B A B . A0 ∨ (A ∧ B0) B0 ∨ (A0 ∧ B) 0 The analogous statement holds for linearly reductive G provided A, A , etc. are all normal in G. For locally compact quantum groups, the quantum analog of Dedekind’s modular law and the Zassenhaus lemma are true only in some cases. The following theorem and proposition summarize the situation.

Theorem 0.4.8 (Theorem 3.3.4 page 91). Let L ≤ H and M be closed quantum subgroups of a locally compact quantum group G such that L normalizes M. Then, we have

H ∧ (M ∨ L) = (H ∧ M) ∨ L. (0.4.2) if either

(a) L is compact, or

(b) H is open in G. 0 0 Proposition 0.4.9 (Proposition 3.3.9 page 93). Let A E A and B E B be either (a) compact or

(b) open quantum subgroups of a locally compact quantum group G. Then, we have an isomorphism 0 ∨ ( ∧ ) 0 ∨ ( ∧ ) A A B =∼ B A B . A0 ∨ (A ∧ B0) B0 ∨ (A0 ∧ B)

0.5 The Schreier reﬁnement theorem and the Jordan-Hölder theorem for quantum groups

In this section, we present the quantum analogue of the Schreier refinement theorem and the Jordan-Hölder theorem for compact and discrete quantum groups (see for example [81, Chapter 5, Theorem 5.11] and [81, Chapter 5, Theorem 5.12]) for the classical analogue for ordinary discrete groups). For this purpose, we first defined the quantum analogue of the notion of subnormal series and composition series. 0.5. The Schreier refinement theorem and the Jordan-Hölder theorem for quantum groups 39

Definition 0.5.1 (Definition 2.3.1 page 81). Let G be either a compact or (algebraic) discrete quantum group. A finite system

G = G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · ≥ Gk = 1 (0.5.1) of closed quantum subgroups of G is called a subnormal series of G if every subgroup Gi is a proper normal closed quantum subgroup of Gi−1 , i ∈ {1, 2, ··· , k}. In particular, G1 is a normal closed quantum subgroup of G, G2 is a normal closed quantum subgroup of G1, but not necessarily of G, and so on. A subnormal series is normal if each Gi is normal in the ambient group G. The corresponding subquotient quantum groups

G1\G, G2\G1, ··· , Gk\Gk−1 of G are the factors of the (sub)normal series (0.5.1). The integer k is the length of the series (0.5.1). Deﬁnition 0.5.2. A subnormal series

G = H0 ≥ H1 ≥ H2 ≥ H3 ≥ · · · ≥ Hl = 1 (0.5.2) is called a refinement of the subnormal series (0.5.1) if every quantum subgroup Gi of (0.5.1) coincides with one of the quantum subgroups Hj, i.e. if every quantum subgroup that occurs in (0.5.1) also occurs in (0.5.2). In particular, every subnormal series is a refinement of itself. The lengths of the normal series (0.5.1) and its refinement (0.5.2) of course satisfy the inequality k ≤ l. Two subnormal series of a compact quantum groups are called equivalent if their lengths are equal and their constituent subquotients are isomorphic up to permutation. We are now ready to state the quantum analogue of the Schreier refinement theorem and the Jordan-Hölder theorem, and their proofs follow respectively from the Zassenhaus lemma and Schreier refinement theorem. Theorem 0.5.3 (Theorem 2.3.3 page 82). Any two subnormal series of a compact / discrete quantum group G have equivalent refinements. The same holds for any two normal series of a linearly reductive quantum group. Definition 0.5.4 (Definition 2.4.1 page 82). A subnormal series (0.5.1) is a composition series of G if Gi is a proper maximal normal closed quantum subgroup of Gi−1 for 1 ≤ i ≤ k. Theorem 0.5.5 (Theorem 2.4.3 page 82). Any two composition series of a compact or discrete quantum group G are equivalent. For locally compact quantum groups, the quantum analog of the Schreier refinement theorem and the Jordan-Hölder theorem are also true in some cases. Let us first note that the Definition 0.5.1, Definition 0.5.2 and Definition 0.5.4 remain valid for locally compact quantum groups. The following two theorems summarize the situation. We write {G`}`≥0 for the generic (sub)normal series

G = G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · ≥ Gk = 1. (0.5.3) of closed quantum subgroups of a locally compact quantum group G. 40 Contents

Theorem 0.5.6 (Theorem 3.4.1 page 94). Let G be a locally compact quantum group. 0 Then, any two subnormal series {G`} and {Gt} of G admit equivalent reﬁnements, provided

0 G`, ` ≥ 1 and Gt, t ≥ 1 are all

(a) compact or

(b) open.

Theorem 0.5.7 (Theorem 3.4.2 page 94). Let G be a locally compact quantum group. Then, all composition series of G consisting of all (a) compact or all

(b) open quantum subgroups are equivalent.

The compact versions of Theorem 0.5.6 and Theorem 0.5.7 refer to subnormal series (0.5.3) in which all G`, ` ≥ 1 are compact, but G = G0 need not be so. Let us note that this is equivalent to the subquotient G/G1 being non-compact. Indeed, we have

Proposition 0.5.8 (Proposition 3.4.3 page 95). A locally compact quantum group G is compact if and only if it admits a subnormal series (3.4.1) with compact quotients Gi/Gi+1.

0.6 Characterization of ergodic actions and idempotents states of the compact quantum group O−1(2)

An idempotent measure on a classical Hausdorff compact group is the Haar measure of a closed subgroup. Pal showed in [78] that an analogue result is no longer possible in the quantum case by giving the example of an idempotent measure on the Kac-Paljutkin compact quantum group of dimension 8 that does not result from the Haar measure of a quantum subgroup. In this section, we first define a compact quantum group before we announce the result of Franz-Skalski[39], who first established a characterization of the idempotent states of a compact quantum group. This result was generalized for locally compact quantum groups by Salmi-Skalski[85].

In this section we will give a complete list of the ergodic actions and embedabble ergodic actions of the compact quantum group O−1(2) and characterize the idempotent states of the latter relying on the result of Franz- Skalski [39, Theorem 4.1]. First, let us note that listing the ergodic actions of the compact quantum group O−1(2) is equivalent to listing the ergodic actions of the classical compact group O(2). Indeed, a fundamental result of Banica-Bichon-Collins, more precisely [6, Theorem 4.3], stipulates that the category of corepresentations of the compact quantum group O−1(n) is tensor equivalent to the category of representation of the classical compact group O(n). From this we deduce that the compact quantum groups O−1(2) and O(2) (which is also a compact quantum group with q = 1) are monoidally equivalent. Another fundamental result of De Rijt-Vander Vennet, more precisely [30, Theorem 7.3], speciﬁes that the ergodic actions of two monoidally equivalent compact quantum groups are in bijective correspondence. We deduce from this that the ergodic actions 0.6. Characterization of ergodic actions and idempotents states of the compact quantum group O−1(2) 41

of the compact quantum group O−1(2) are in bijective correspondence with the ergodic actions of the classical compact group O(2). The first difficulty was to list the ergodic actions of the classical compact group O(2). Let us first remark that if G is a classical compact group, H a closed subgroup, and α H y N an ergodic action of H on a von Neumann algebra N, we have the induced ergodic action of G on

G ∞ IndH (N) = {f ∈ L (G, N) | ∀g ∈ G, h ∈ H, αh(f(gh)) = f(g)} ⊆ L∞(G)⊗¯ N, given by 0 −1 0 (αg(f))(g ) = f(g g ).

In particular, if π is an irreducible projective representation of H on a Hilbert space Vπ, G we have an ergodic action of G on the von Neumann algebra IndH (B(Vπ)) where B(Vπ) is equipped with the ergodic H-action

∗ αh(x) = π(h)xπ(h) .

For example, when π is the trivial representation (or a one-dimensional representation), G ∞ we have IndH (C) = L (G/H) with the left translation action. ∼ In the following, we will identify O(2) = C2 n T , with T = {z ∈ C | |z| = 1} the circle group, and with the cyclic group C2 = {1, σ} acting on T by σ(z) =z ¯.

The following result is a private communication from Kenny De Commer.

Theorem 0.6.1 (Theorem 4.1.2 page 98). Let O(2) M be an ergodic action. Then yα ∼ O(2) M = IndH (B(Vπ)) for some closed subgroup H ⊆ O(2) and π an irreducible representation of H.

After having determined the isomorphisms between the induced actions, we have obtained the complete list of the non-equivalent ergodic actions of the classical compact group O(2). (∞) The irreducible (projective) representations of O(2) give either the trivial action β0 (∞) on C (for the characters) or the ergodic action βl/2 = α on M2(C) by ! ! ! ! a b a zlb a b d c α = , α = z c d z−lc d σ c d b a for l ∈ N \{0} (with the even l coming from representations and the odd l coming from non-trivial projective representations). (∞) 2 The irreducible representations of T all give the same action α = α on C = L∞(O(2)/T ), namely

αz((x, y)) = (x, y), ασ(x, y) = (y, x).

(k) The irreducible representations of Ck for some ﬁxed k all give the same action α = α ∞ ∞ ∞ on L (O(2)/Ck) = L (T/Ck) ⊕ L (T/Ck), namely

αz(f, g) = (fz, gz), ασ(f, g) = (g, f), where fz denotes the z-translate of f. 42 Contents

(k) Finally, for the dihedral group Dk we have the action α = β0 coming from the ∞ characters of Dk, giving the action on L (O(2)/Dk), or for the non-character (projective) (k) (k) representations the actions βl/2 where 0 < l < k a natural number and βl/2 the induced action coming from the (projective) Dk-representation ! ! ! ! a b a zlb a b d c α = , α = . z c d z−lc d σ c d b a

Hence, we obtain: Proposition 0.6.2 (Proposition 4.1.3 page 100). The full list of non-equivalent ergodic actions of O(2):

(k) (∞) (∞) (k0) 0 0 {βl/2 , βl0/2 , α , α | k, k ∈ N0, l ∈ N, 0 ≤ l < k}.

We want to deduce the list of the embeddable ergodic actions of O−1(2). To do this, we proceed as follows. First, note that by definition an embeddable ergodic action is a comodule ∗-algebra of the CQG algebra A−1 associated to O−1(2) which embeds into A−1 as such (i.e. by an embedding that preserves all of the structure: comodule, algebra, etc.). Since the twisting equivalence λ. (see the notation from Section 4.4) that implements Theorem 1.4.19 also implements an equivalence between the categories of coideal ∗-algebras over A−1 and the untwisted version A (algebra of representative functions on the classical group O(2)), it will be sufficient to identify the ergodic O(2)-action B in the list of Proposition 0.6.2 for which ∼ λ . B = A−1 as A−1 comodule ∗-algebras, and to then also identify the members of that list that embed into B. (2) We will see that there is only one candidate for B (namely β1/2) using the Peter- Weyl theorem to determine the representation type of the ergodic actions identified in Proposition 0.6.2 (where by representation type we mean the multiplicities of the various irreducible O(2)-representations). Indeed, this is the substance of the following result. Proposition 0.6.3 (Proposition 4.2.1 page 100). The only comodule algebras among those in Proposition 0.6.2 that are isomorphic to A as O(2)-representations are α(1) =∼ A (2) itself and β1/2. We then obtained as a consequence the complete list of the embeddable ergodic coactions of the compact quantum group O−1(2). Corollary 0.6.4 (Corollary 4.2.2 page 101). The twisting equivalence λ. induces a bijection between (k) (k) {α , βl/2 | k = ∞ or even , l = 0 or odd } from Proposition 0.6.2 and the isomorphism classes of embeddable ergodic actions of O−1(2).

The complete list of non-equivalent embeddable ergodic coactions of O−1(2) is then:

(k) (k) {α , βl/2 | k = ∞ or even , l = 0 or odd }

To ﬁnish this section, we asked ourselves the following question motivated by Theo- rem 1.4.19 page 71: 0.6. Characterization of ergodic actions and idempotents states of the compact quantum group O−1(2) 43

Question 0.6.1 (Question 1.4.1 page 71). Let G1 and G2 be two monoidally equivalent compact quantum groups. Is there a bijective correspondence between their embeddable ergodic actions?

Remark 0.6.5. An aﬃrmative answer to the previous question would extend the fundamental result [30, Theorem 7.3] of De Rijt-Vander Vennet to embeddable ergodic actions. But the answer is negative in general (See Question 0.6.2 below) except in the case we have an additional condition: See Proposition 4.4.5 page 106).

Speciﬁcally, let G1 and G2 be two monoidally equivalent compact quantum groups. Suppose furthermore that the monoidal equivalence is implemented by twisting by a cocycle ⊗2 λ : C(G1) → C. Then, according to the proof of Theorem 1.4.19, we have a twisting operation λ. that implements an equivalence between the category Erg(G1) of ergodic G1-actions and the analogous category Erg(G2). In this setting, the formulation of Question 0.6.1 that we answer here is

Question 0.6.2. Does λ. : Erg(G1) → Erg(G2) restrict to an equivalence between subcategories of embeddable ergodic coactions?

Unfortunately, the answer to Question 0.6.2 is negative in general, as shown in the counter-example below. Take the compact quantum group G1 to be the order-64 Heisenberg group

4 4 4 H4 := hε1, ε2, δ | ε1 = ε2 = δ = 1, δε1 = ε1δ, ε2δ = δε2, ε1ε2 = δε2ε1i.

Its deformation G2 (with underlying function algebra H = C(H4)) will be the one used in [16], described in some detail in Section 6 of [16]. Although we will not write the cocycle

⊗ λ : C(H4) → C explicitly, recall that it is obtained as the composition

λ ⊗2 C(H4) C C(Γ)⊗2 where the left hand arrow is restriction along the inclusion

2 ∼ 2 2 (Z/2) = Γ := hε1, ε2i ⊂ H4 and the right hand arrow is some cocycle inducing the non-zero cohomology class in

2 × ∼ H (Γb, C ) = Z/2.

It will be more convenient to dualize and work with the group algebra CH4 and its ∗ deformed version H , which coincides with CH4 as an algebra but has twisted comultiplication. We proved that: 44 Contents

Proposition 0.6.6 (Corollary 4.4.3 page 106). There does not exist an inclusion, dimension, and G-preserving bijection between the coideal subalgebras of CH4 and those of H∗.

As one ﬁnal simple observation, we remark that if G1 happens to have trivial 2- 2 1 cohomology set H (Gc1, S ), the answer to Question 0.6.2 is aﬃrmative:

Proposition 0.6.7 (Proposition 4.4.5 page 106). Let G1 and G2 be two monoidally equiv- 2 1 alent compact quantum groups, and assume H (Gc2, S ) is trivial. Then,

λ. : Erg(G1) → Erg(G2) restricts to an equivalence between subcategories of embeddable ergodic actions.

Idempotent states of the compact quantum group O−1(2)

Having succeeded to make the list of the embeddable ergodic actions of O−1(2) and since by definition, an embeddable ergodic action is identified with a coideal subalgebra and since according to the fundamental result [39, Theorem 4.1] by Franz-Skalski, there is a bijective correspondence between the expected coideal subalgebras and the idempotent states of a compact quantum group; Our project has met a final concern: How to express by a concrete formula an idempotent state from an action of a compact quantum group? In fact, although we characterized them by listing embeddable ergodic action using [39], we still have to list explicitly the idempotent states of O−1(2) from the list of its embeddable ergodic actions. Considering that:

1. Let us recall that if one takes the comultiplication itself as the action (i.e the quantum group itself is taken as the coideal subalgebra), it is not immediate to produce the corresponding Haar state from the multiplication.

2. Also if we consider the subalgebra C, then we see that we can not produce the Haar state from the action which is simply C → C ⊗ C, 1 7→ 1 ⊗ 1. The problem is that starting from the proof of [39, Theorem 4.1], there are several elements that we have abstractly but non-explicitly, and they depend on each other. Indeed, to obtain the idempotent state of the coideal subalgebra, one needs the conditional expectation E; And to get E, we need unfortunately the state in the proof of [39, Theorem 4.1]. It is therefore circular and does not lead to something explicit (just to the theorem of the paper of Franz-Skalski, i.e the fact that there is this bijection). We believe that if there is a chance to obtain a concrete formula of the idempotent state from an embeddable ergodic action, it will be by expressing the conditional expectation E only as a function of the embeddable ergodic action α by not involving the state itself, then compose E with the counit ε.

The following question summarizes our point of view.

Question 0.6.3 (Question 4.3.1 page 104). Is it possible to obtain such a concrete formula in general for a compact quantum group at least in the Kac case? Chapter 1

Preliminaries

This chapter collects the necessary preliminaries for the whole thesis. The ﬁrst section is a brief introduction to locally compact quantum group (in the von Neumann algebra setting). In the second section we present some preliminaries on linearly reductive compact quantum groups, including some basic notions and properties, as well as some typical constructions such as the free products and the Drinfeld-Jimbo deformation.

1.1 Preliminaries for locally compact quantum groups

The theory of locally compact quantum groups is formulated in terms of operator algebras. Operator algebra theory is divided into two parts. In order to explain this division let us ﬁx a Hilbert space H. The set of all bounded operators acting on H forms a normed ∗- algebra which we denote by B(H). This algebra, except the norm topology carries a host of locally convex topologies: strong, σ-strong, weak, σ-weak and others. Although the aforementioned distinction does not depend on the choice of the topology listed above, we choose the σ-weak topology on B(H) for its description. In this paper the scalar product (·|·) on a Hilbert space will be linear in the second variable.

Deﬁnition 1.1.1. Let I be a directed set and H a Hilbert space. Let (Ti)i∈I be a net of bounded operators acting on H and let T ∈ B(H). We say that (Ti)i∈I σ-weakly converges ∞ ∞ X 2 X 2 to T ∈ B(H) if for all sequences (ξn)n∈N, (ζn)n∈N ∈ H satisfying kξnk < ∞, kζnk < n=1 n=1 ∞ we have ∞ ∞ X X lim (ξn|Tiζn) = (ξn|T ζn). i n=1 n=1

We say that (Ti)i∈I σ-∗ strongly converges to T if

∞ X 2 lim k(T − Ti)ζnk =0, i n=1 ∞ X ∗ ∗ 2 lim k(T − Ti )ζnk =0. i n=1 Deﬁnition 1.1.2. Let H be a Hilbert space.

(i)A ∗-subalgebra A of B(H) which is closed in the norm topology is called a concrete C∗-algebra. 46 Chapter 1. Preliminaries

(ii) A unital ∗-subalgebra N of B(H) which is closed in the σ-weak topology is called von Neumann algebra.

Usually we shall skip the term concrete and say that A ⊂ B(H) is a C∗-algebra. Let X ⊂ B(H) be a non-empty subset. The commutant X0 of X is deﬁned as

X0 = {y ∈ B(H): xy = yx for all x ∈ X}.

We shall write X00 = (X0)0. The famous bicommutant theorem implies that a *-subalgebra A ⊂ B(H) is a von Neumann algebra if and only if A = (A0)0. Let Y be a topological vector space and X a non-empty subset of Y. The closure of the linear span of X will be denoted Xcls. If X is a C∗-algebra then the norm closure of the linear span of X will also be denoted by Xk·k−cls. If X is a von Neumann algebra then the σ-weak closure of the linear span of X will be denoted by Xσ−cls. ∗ Given a pair of C -algebras A1 ⊂ B(H1) and A2 ⊂ B(H2), the (spatial) tensor product A1 ⊗ A2 ⊂ B(H1 ⊗ H2) is deﬁned as

k·k−cls A1 ⊗ A2 = {x ⊗ y : x ∈ A1, y ∈ A2} .

Similarly, given a pair of von Neumann algebras N1 ⊂ B(H1) and N2 ⊂ B(H2), we deﬁne

σ−cls N1 ⊗¯ N2 = {x ⊗ y : x ∈ N1, y ∈ N2} .

The Banach dual of the Banach space (B(H), k · k) will be denoted by B(H)∗. For ∗ ζ, ξ ∈ H we deﬁne a bounded functional ωζ,ξ ∈ B(H) : ωζ,ξ(T ) = (ζ|T ξ) for all T ∈ B(H). Let us consider a subset X of B(H)∗:

X = {ωζ,ξ : ζ, ξ ∈ H}.

k·k−cls We shall denote B(H)∗ = X . We say that ω ∈ B(H)∗ is a normal functional on B(H). ∗ We have (B(H)∗) = B(H) and the σ-weak topology coincide with the weak ∗-topology on B(H). There is an abstract version of a (concrete) concept of a C∗-algebra and a von Neumann algebra formulated in Deﬁnition 1.1.2.

Deﬁnition 1.1.3. Let A be a Banach ∗-algebra. We say that A is a C∗-algebra if the C∗-identity ka∗ak = kak2 holds for all a ∈ A. Let N be a C∗-algebra. We say that N is a W ∗-algebra if N admits a predual Banach space.

Every C∗-algebra can be identiﬁed with a concrete C∗-algebra. A C∗-algebra N can be identiﬁed with a von Neumann algebra if and only if N is a W ∗-algebra. The predual ∗ space of a W -algebra N is uniquely determined by N and it will be denoted by N∗ In this paper we shall always consider concrete C∗-algebras which are non-degenerate.

Deﬁnition 1.1.4. Let A ⊂ B(H) be a concrete C∗-algebra. We say that A is non- T degenerate if a∈A ker a = {0}. Let A ⊂ B(H) be a non-degenerate C∗-algebra. The C∗-algebra

M(A) = {x ∈ B(H): xa, ax ∈ A} is called a multiplier C∗-algebra of A. 1.1. Preliminaries for locally compact quantum groups 47

It can be proved that the multiplier C∗-algebra M(A) of a concrete C∗-algebra A ⊂ B(H) does not depend on the embedding A ⊂ B(H). To be more precise if K is a Hilbert space and π : A → B(K) is an injective ∗-homomorphism then π(A) is a C∗-subalgebra of B(K) and M(A) and M(π(A)) are isomorphic (as C∗-algebras). Let B be a C∗-algebra and C a C∗-subalgebra of M(B). The set

{cb : c ∈ C, b ∈ B}k·k−cls will be denoted CB. Let π : A → M(B) be a ∗-homomorphism. We say that π is non- degenerate if π(A)B = B. The set of non-degenerate ∗-homomorphisms from A to M(B) will be denoted by Mor(A, B). It can be checked that there exists a unique ∗-homomorphism π : M(A) → M(B) such that for all x ∈ M(A) and a ∈ A we have π(xa) = π(x)π(a). In particular π extends π and in what follows this extension will be denoted by π. Note that for π ∈ Mor(A, B) and ρ ∈ Mor(B, C) we can form ρ ◦ π ∈ Mor(A, C). This composition gives rise to the category of C∗-algebras with Mor(A, B) being morphisms. Let N and M be von Neumann algebras. A unital ∗-homomorphism π : N → M is said to be normal if it is continuous in the σ-weak topologies. The set of positive elements of N will be denoted by N+. Deﬁnition 1.1.5. Let M be a von Neumann algebra. A weight on M is a function + ψ : M → R≥0 ∪ {∞} such that ψ(0) = 0, ψ(x + y) = ψ(x) + ψ(y) and ψ(tx) = tψ(x) for + all t ∈ R≥0 and x, y ∈ M . We say that ψ is normal if it is lower semi-continuous in the σ-weak topology on M+. We say that ψ is semiﬁnite if the set

{x ∈ M+ : ψ(x) < ∞} is σ-weakly dense in M+. We say that that ψ is faithful if ψ(x) = 0 =⇒ x = 0. A normal semiﬁnite faithful weight will be called an n.s.f. weight. Let ψ be an n.s.f. weight on M. Then we deﬁne the following sets:

+ + •M ψ = {x ∈ M : ψ(x) < ∞}, ∗ •N ψ = {x ∈ M : ψ(x x) < ∞}, + •M ψ = LinMψ . ∗ Let us note that Nψ forms a left ideal in N. It can be checked that Mψ = Lin{x y : x, y ∈ Nψ} and ψ yields a linear map ψ : Mψ → C. The GNS-construction based on ψ is a triple (Hψ, πψ, η) where Hψ is a Hilbert space πψ : N → B(Hψ) is a normal ∗-homomorphism and η : Nψ → Hψ is a σ-∗ strongly closed linear map such that

∗ • (η(x)|η(y)) = ψ(x y) for all x, y ∈ Nψ,

• η(xy) = πψ(x)η(y) for all x ∈ N and y ∈ Nψ. A GNS construction for ψ always exists and is essentially unique. For the theory of locally compact quantum groups we refer to [63, 65]. Deﬁnition 1.1.6. A von Neumann algebraic locally compact quantum group is a quadru- ¯ ple G = (M, ∆G, ϕG, ψG), where M is a von Neumann algebra, ∆ : M → M ⊗ M is a normal injective ∗-homomorphism satisfying

(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ 48 Chapter 1. Preliminaries

and ϕG and ψG are, respectively, normal semiﬁnite faithful left and right invariant weights on M, i.e.

ψG((id ⊗ ω)(∆(x)) = ψG(x)

ϕG((ω ⊗ id)(∆(x)) = ϕG(x) + + for all x ∈ M and ω ∈ M∗ . ∞ Let G be a locally compact quantum group. We shall use a notation M = L (G), ∆ = 2 ∆G. The GNS Hilbert space of the right Haar weight ψG will be denoted by L (G) and the corresponding GNS map will be denoted by ηG. G is assigned with the antipode, the scaling group and the unitary antipode which are denoted by S, (τt)t∈R and R. A fundamental role in the theory of locally compact quantum groups is played by the multiplicative unitary 2 2 WG ∈ B(L (G) ⊗ L (G)), which is a unique unitary operator such that

G 1 W (ηG(x) ⊗ ηG(y)) = (ηG ⊗ ηG)(∆G(x)( ⊗ y))

G G G G G G for all x, y ∈ D(ηG); W satisﬁes the pentagonal equation W12W13W23 = W23W12. Note G G 1 2 2 2 that we use the leg numbering notation, e.g. W12 = W ⊗ ∈ B(L (G) ⊗ L (G) ⊗ L (G)). G ∞ Using W one can recover L (G) and ∆G

∞ G 2 σ−cls L (G) = (ω ⊗ id)(W ) ω ∈ B(L (G))∗ , ∗ G 1 G ∆G(x) =W (x ⊗ )W . ∗ A locally compact quantum group G admits a C -version, which can also be recovered ∗ from WG. For example the C -algebra assigned to G denoted by C0(G) is given by

G 2 k·k−cls C0(G) = (ω ⊗ id)(W ) ω ∈ B(L (G))∗ .

We say that G is a compact quantum group if 1 ∈ C0(G). A locally compact quantum ∞ group admits a dual object Gb = (L (Gb ), ∆ , ϕ , ψ ). For the detailed description of the Gb Gb Gb 2 2 Haar weights ϕ , ψ we refer to [63]; let us only mention that we have L (Gb ) = L (G). Gb Gb ∗ The multiplicative unitary assigned to Gb is given by WGb = σ(WG) , where σ denotes the ﬂipping morphism σ(a ⊗ b) = b ⊗ a. In particular we have

∞ G 2 σ−cls L (Gb ) = (ω ⊗ id)(Wb) ω ∈ B(L (G))∗ , ∗ ∆ (x) = WGb(x ⊗ 1)WGb . Gb Moreover G 2 k·k−cls C0(Gb ) = (ω ⊗ id)(Wb) ω ∈ B(L (G))∗ and we have WG ∈ M(C0(Gb ) ⊗ C0(G)). Definition 1.1.7. Let G be a locally compact quantum group. The opposite locally compact quantum group op is defined as (L∞( ), ∆op, ψ , ϕ ) where ∆op = σ ◦ ∆ . We G G G G G G G say that G is abelian if ∆G = ∆Gop ; in other words G is abelian if and only if Gb is classical, ∞ i.e. L (Gb ) is commutative. Definition 1.1.8. Let G be a locally compact quantum group, N a von Neumann algebra ∞ and α : N → L (G) ⊗¯ N a normal, unital injective ∗-homomorphism. We say that α is a left action of G on N if (∆G ⊗ id) ◦ α = (id ⊗ α) ◦ α. 1.1. Preliminaries for locally compact quantum groups 49

We say that the action α is integrable if the set

+ + {x ∈ N :(ψG ⊗ id)(α(x)) ∈ N } is σ-weakly dense in N+.

For an action α of G on N we have (see [59, Corollary 2.6])

∞ σ−cls N = {(ω ⊗ id)(α(x)) : ω ∈ L (G)∗, x ∈ N} . (1.1.1)

We also have a right counterpart of the concept of an action and the integrability condition. ∞ ∞ ∞ In the sequel we shall often use the right adjoint action β :L (Gb ) → L (Gb ) ⊗¯ L (G) of on L∞( b ) where G G ∗ β(x) = WG(x ⊗ 1)WG (1.1.2) ∞ for all x ∈ L (Gb ). ∞ A von Neumann subalgebra N ⊂ L (G) is called ∞ ¯ • Left coideal if ∆G(N) ⊂ L (G) ⊗ N; ¯ • Invariant subalgebra if ∆G(N) ⊂ N ⊗ N; ∞ • Baaj-Vaes subalgebra if N is an invariant subalgebra of L (G) which is preserved by unitary antipode R and the scaling group (τt)t∈R of G; ∗ ∞ • Normal if WG(1 ⊗ N)WG ⊂ L (Gb ) ⊗¯ N;

• Integrable if the set of integrable elements with respect to the right Haar weight ψG + is dense in N ; in other words, the restriction of ψG to N is semiﬁnite. ∞ In the sequel a von Neumann subalgebra of L (G) which is a left coideal will be called a G-coideal or simply a coideal. Note that ∆G|N is an action of G on N. In particular (see (1.1.1)) ∞ σ−cls N = {(ω ⊗ id)(∆G(x)) : ω ∈ L (G)∗, x ∈ N} . (1.1.3) ∞ Let N ⊂ L (G) be a Baaj-Vaes subalgebra. The restriction of ∆G to N will be denoted by ∆N : N → N ⊗¯ N. We shall often use the so called Baaj-Vaes theorem [98, Proposi- tion 10.5], which states that (N, ∆|N) admits a structure of a locally compact quantum group. To be more precise there exists a pair of n.s.f. weights ϕN, ψN on N such that (N, ∆G|N, ϕN, ψN) is a locally compact quantum group.

Deﬁnition 1.1.9. Let G be a locally compact quantum group. The set of G-coideals will be denoted by CI(G). We equip CI(G) with the poset structure: for N, M ∈ CI(G) we write N ≤ M if N ⊂ M. The poset (CI(G), ≤) admits two operations ∧, ∨ • N ∧ M = N ∩ M,

• N ∨ M = {xy : x ∈ N, y ∈ M}00.

(CI(G), ≤, ∧, ∨) forms a lattice which will be called a lattice of coideals of G. The subset of CI(G) of normal G-coideals will be denoted NCI(G). The subset of ∞ CI(G) of Baaj-Vaes subalgebras of L (G) will be denoted BV(G). It is easy to check that NCI(G) and BV(G) form sublattices of CI(G). Similarly NCI(G) ∩ BV(G) forms a sublattice of CI(G). 50 Chapter 1. Preliminaries

Remark 1.1.10. Using [55, Theorem 3.9] we get a bijective map CI(G) 3 N 7→ N˜ ∈ CI(Gb ) where 0 ∞ N˜ = N ∩ L (Gb ). The coideal N˜ is said to be a codual of N and the map N 7→ N˜ is denoted by cd. Note that cd : CI(G) → CI(Gb ) is an anti-isomorphism of lattices: cd(N) ≤ cd(M) iﬀ M ≤ N, cd(N ∧ M) = cd(N) ∨ cd(M), cd(N ∨ M) = cd(N) ∧ cd(M).

2 Moreover cd = id (note that the coduality CI(Gb ) → CI(G)) is also denoted by cd).

1.2 Preliminaries for linearly reductive quantum groups

In Chapter2 we work with Hopf algebras over an algebraically closed field k, regarded as the function algebras of the quantum groups in question. For this reason, we typically speak of either the quantum group G or the Hopf algebra O(G) associated to it. G is then a non-commutative analogue of a linear algebraic group [9]. Unless specified otherwise, antipodes are assumed to be bijective. For general background on coalgebras, bialgebras or Hopf algebras (which we recall somewhat briefly and selectively) the reader may consult e.g. [1, 27, 80, 92]; the various papers we cite are also good sources on specific points that arise in the course of the discussion below. We use Sweedler notation for the comultiplication of a Hopf algebra (or more generally coalgebra) H, writing ∆ : x 7→ x1 ⊗ x2 for the comultiplication ∆ : H → H ⊗ H. The reader should note that the symbol ⊗ has double meaning in this thesis - one in the context of C∗-algebras and other in the context of algebras. Counits and antipodes are denoted by ε and S. Finally, for a linear subspace V ⊆ H of H, we denote − V := ker(ε|V ).

We denote categories of left / right modules over an algebra A by AM and MA respectively. Similarly, the categories of left / right C-comodules for a coalgebra C are denoted by C M and MC respectively. Following standard terminology (see e.g. [73, Definition 1.4]), the quantum group G is linearly reductive when O(G) is cosemisimple, i.e. its category MO(G) is semisimple. Every coalgebra is the union of its finite-dimensional subcoalgebras (the so-called fundamental theorem of coalgebras; e.g. [92, Theorem 2.2.1]), and cosemisimple coalgebras are direct sums of their simple subcoalgebras, i.e. those that have no proper non-zero subcoalgebras [92, §14]. This latter decomposition is dual to the usual decomposition of semisimple algebras as (finite) products of simple algebras. In fact, simple coalgebras are dual to simple algebras, and hence, since we are working over an algebraically closed field, ∗ all simple subcoalgebras are of the form Mn (duals of the matrix algebras Mn = Mn(k)). We will also deal with discrete quantum groups in a slightly more general setting than in Chapter3. Definition 1.2.1, summarizing our conventions, will be sufficient for our purposes.

Definition 1.2.1. The category QG of (linear algebraic) quantum groups over a fixed field k is the category opposite to that of Hopf algebras over k with bijective antipode. 1.2. Preliminaries for linearly reductive quantum groups 51

The category RQG of linearly reductive quantum groups over a fixed field k is the full subcategory of QG consisting of cosemisimple Hopf algebras. The category DQG of algebraic discrete quantum groups over a fixed field k is the opposite category RQGop. We often drop the adjective ‘algebraic’ below.

Remark 1.2.2. In other words, we regard discrete quantum groups as dual to linearly reductive groups. This mimics the usual machinery in the locally compact case, except that we allow here arbitrary algebraically closed ﬁelds of arbitrary characteristic, and there are no ∗ structures.

In the spirit of Deﬁnition 1.2.1, we regard the underlying Hopf algebra O(Gb ) of a linearly reductive quantum group Gb as the group algebra of its discrete Pontryagin dual G of Gb and (working over the algebraically closed ﬁeld k) use the notation

kG = O(Gb ) when we want to emphasize this point of view. One particular class of cosemisimple Hopf algebras are the CQG algebras of [32], which ∞ in the context of Section 1.1 are dense complex Hopf ∗-subalgebras of L (G) for some compact quantum group G. Some of the results of Chapter2 only apply to CQG algebras.

Deﬁnition 1.2.3. A quantum subgroup of the linearly reductive quantum group G is a quotient Hopf algebra O(G) → O(K). Let G be a discrete quantum group. A quantum subgroup of G is a Hopf subalgebra A ⊆ kG. Remark 1.2.4. Note that quantum subgroups of a discrete quantum group are automatically discrete, because cosemisimplicity is preserved by passing to Hopf subalgebras. Thus denoting A = kK in the second part of Deﬁnition 1.2.3 we see that a quantum subgroup K of a discrete quantum group G is a Hopf subalgebra kK ⊆ kG. A quantum subgroups of linearly reductive quantum groups need not be reductive, however: consider the classical situation whereby the function algebra of GL2(C) surjects onto that of the subgroup of upper triangular invertible matrices.

Quantum subgroups of a given quantum group form a lattice both in QG and in DQG.

Deﬁnition 1.2.5. Let O(G) → O(Hi), i = 1, 2 be two quantum subgroups of G ∈ QG. Then, the intersection H1 ∧ H2 whose underlying Hopf algebra O(H1 ∧ H2) is deﬁned as −1 the quotient of O(G) by the smallest Hopf ideal I invariant under the inverse S of the antipode of G, and which contains both ideals

ker (O(G) → O(Hi)) , i = 1, 2. (1.2.1)

Similarly, the subgroup H1 ∨ H2 generated by Hi is deﬁned as the object in QG whose underlying Hopf algebra is the quotient of O(G) by the largest Hopf ideal invariant under S−1 contained in both (1.2.1). Now let kHi ⊆ kG be two quantum subgroups of a discrete quantum group G. Then, the intersection H1 ∧ H2 is the discrete quantum group whose underlying group algebra k(H1 ∧ H2) is the intersection of kHi in kG. Similarly, the subgroup generated by Hi is deﬁned as the discrete quantum group whose underlying Hopf algebra is the Hopf subalgebra of kG generated as an algebra by kHi. 52 Chapter 1. Preliminaries

We can verify that in both cases the operations ∧ and ∨ are well deﬁned and turn the sets of quantum subgroups into lattices.

Remark 1.2.6. Classically, the intersection H ∧ K can be deﬁned as the pullback

H H ∧ K G K in whatever category of groups is convenient (linear algebraic, etc.). The analogue in the QG case of Deﬁnition 1.2.5 is the observation that we have a pushout

O(H) O(H ∧ K) O(G) O(K) in the category of algebras, or equivalently, that of Hopf algebras (or Hopf algebras with bijective antipode). In other words, the left hand corner is universal among quotients that make the diagram commutative. We will make frequent (mostly implicit) use of an algebraic version of the correspondence cd from Section 1.1 throughout Chapter2. We elaborate on the construction here. First, for any Hopf algebra H, define CI(H) as the set of right coideal subalgebras A ⊆ H, i.e. those subalgebras for which ∆(A) ⊆ A ⊗ H. (in opposition to Section 1.1, we use right rather than left coideals in order to preserve compatibility with much of the literature on Hopf algebras accessible through our references). Now, for each A ⊆ H in CI(H) denote cd(A) = H/HA−. This is a left module quotient coalgebra of H in the sense of Definition 1.2.7 (e.g. [94, Proposition 1]), which justifies denoting the set of such module quotients by MQ(H). Definition 1.2.7. A (left) module coalgebra over a Hopf algebra H is a coalgebra C equipped with an H-module action H ⊗ C → C that is a coalgebra map. A (left) module quotient coalgebra is a module coalgebra H ⊗C → C as before equipped with a surjection H → C of module coalgebras, i.e. a surjection that is both a coalgebra morphism and a morphism of left H-modules. On the other hand, given π : H → C in MQ(H), define cd(π) (or usually cd(C) by a slight abuse of notation) to be

cd(π) := {x ∈ H | π(x1) ⊗ x2 = π(1) ⊗ x}. It can be shown to be an object in CI(H) ([94, Proposition 1]). Note that we are using the symbol cd for two diﬀerent maps, relating CI and MQ in two opposite directions. They are not, in general, mutual inverses; that requires additional technical conditions, as we now recall. 1.2. Preliminaries for linearly reductive quantum groups 53

Definition 1.2.8. Let ι : A → H be an algebra map. H is left (resp. right) faithfully flat if the functor − ⊗A H : MA → MH resp. H ⊗A − : AM → H M preserves morphism injectivity. Dually, let π : H → C be a coalgebra map. Then, H is left (resp. right) faithfully coflat if the functor C H −C H : M → M resp. C H HC − : M → M preserves morphism surjectivity.

The notion of tensor coproduct , dual to that of tensor product, (see e.g. [94, §1]) can be brieﬂy described as follows: Given a right C-comodule ρV : V → V ⊗ C and a left C-comodule ρW : W → C ⊗ W,

V C W is the subspace of V ⊗ W on which the two arrows

ρV ⊗idW V ⊗ W V ⊗ C ⊗ W idV ⊗ρW agree. Then, part of the content of [94, Theorems 1] is that

cd2 : CI(H) → CI(H) restricts to the identity to those A ∈ CI(H) for which H is left A-faithfully ﬂat. Similarly, [94, Theorem 2] says (among other things) that

cd2 : MQ(H) → MQ(H) restricts to the identity on those π : H → C over which H is right faithfully coﬂat. We will be concerned almost exclusively with situations where either ι : A → H or π : H → C is a Hopf algebra map. For that reason, we make the following simple observation: Lemma 1.2.9. Let π : H → C be a quotient Hopf algebra, and set A = cd(π). Then, A is invariant under the right adjoint action of H on itself deﬁned by

a / x = S(x1)ax2. One consequence of Lemma 1.2.9 that we will use later is Corollary 1.2.10. Let A, B ∈ CI(H) and suppose A = cd(π) for some π : H → C in MQ(C). Then, the linear span

BA = span{ba | a ∈ A, b ∈ B} is a coideal subalgebra. 54 Chapter 1. Preliminaries

Proof. Since it is clear that the space in question is a coideal, it suﬃces to show that it is a subalgebra. Speciﬁcally, we have to prove that for any a ∈ A and b ∈ B, the product ab belongs to BA. This follows from the identity

ab = b1(S(b2)ab3), together with the fact that according to Lemma 1.2.9 the parenthetic factors on the right hand side belong to A.

Let us also record the dual version of Lemma 1.2.9 (the proof is entirely analogous; we once more do not include it):

Lemma 1.2.11. Let ι : A → H be an inclusion of Hopf algebras, and set π : H → C to be cd(A). Then, the left adjoint coaction of H on itself deﬁned by

x 7→ x1S(x3) ⊗ x2 descends to a coaction of H on C through the quotient π : H → C.

We will often encounter the situation when both ι : A → H and π : H → C are morphisms in QG. Given the results recalled briefly above on the importance of (co)flatness, we fix our terminology as follows.

Deﬁnition 1.2.12. An exact sequence of quantum groups is a diagram

k → O(H) → O(G) → O(K) → k (1.2.2) in QG where the second arrow is an inclusion, the third arrow is a surjection, cd inter- changes these two arrows, and moreover O(G) is (co)ﬂat over O(H) (respectively O(K)). In this case we denote H = G/K. The quantum subgroup O(G) → O(K) of G is normal if it ﬁts into an exact sequence (1.2.2). The discrete quantum subgroup kK ⊆ kG of kG ∈ DQG is normal if the inclusion in question is the second arrow in an exact sequence (1.2.2).

Remark 1.2.13. Cf. [3], where the deﬁnition of an exact sequence is the same, minus the (co)ﬂatness conditions.

1.3 Lattice of closed quantum subgroups: basic facts

Deﬁnition 1.3.1. Let G be a locally compact quantum group. The lattice BV(Gb ) will be denoted QS(G) and called a lattice of quantum subgroups of G. The lattice NCI(Gb ) ∩ BV(Gb ) will be denoted N QS(G) and called a lattice of normal quantum subgroups of G.

Let N ∈ QS(G). Using Baaj-Vaes theorem, we conclude the existence of a locally ∞ compact quantum group H such that N = L (Hb ). Thus when convenient we will write H ∈ QS(G). Similarly for H1, H2 ∈ QS(G) we write H1 ∧ H2, H1 ∨ H2.

Remark 1.3.2. Let H ∈ QS(G) and let cd : CI(G) → CI(Gb ) be the coduality (see Remark ∞ ∞ 1.1.10). Then cd(L (Hb )) ∈ CI(G) is denoted by L (G/H). It can be checked that (see e.g. [56]) 1.3. Lattice of closed quantum subgroups: basic facts 55

• cd(QS(G)) ⊂ N CI(G),

• cd(N QS(G)) = N QS(Gb ).

If H ∈ N QS(G) then the normal quantum subgroup cd(H) ∈ N QS(Gb ) is denoted by G[/H. For the concept of short exact sequence of locally compact quantum groups we refer to [99, Deﬁnition 3.2.]. Up to natural isomorphisms all examples are of the form

• → H → G → G/H → • (1.3.1) where • denotes a trivial group. Since cd2 = id we also have the dual exact sequence

• → G[/H → Gb → Hb → •. (1.3.2)

Let G be a locally compact quantum group. As formulated in Deﬁnition 1.3.1, a closed ∞ quantum subgroup of G corresponds to a Baaj-Vaes subalgebra of L (Gb ). In particular a locally compact quantum group G can be assigned with a quantum subgroup Z(G) ≤ G ∞ which is called a center of G: by deﬁnition L (Z\(G)) is the largest Baaj-Vaes subalgebra ∞ contained in the center of the von Neumann algebra L (Gb ). In particular Z(G) is a normal quantum subgroup of G and one can form the quotient group G/Z(G). For the detailed description of the corresponding exact sequence

• → Z(G) → G → G/Z(G) → • see [54]. In what follows we shall describe the quantum analog of the quotient of G by its commutator subgroup.

Proposition 1.3.3. Let G be a locally compact quantum group and let us consider M = {x ∈ L∞( ):(id ⊗ ∆op)(∆ (x)) = (id ⊗ ∆ )(∆ (x))}. (1.3.3) G G G G G ∞ ∞ Then M is a Baaj-Vaes subalgebra of L (G). The quantum group H such that M = L (H) is abelian. Let N be another Baaj-Vaes subalgebra and L be the locally compact quantum group assigned to N. If L is abelian then N ⊂ M. ∞ Proof. Clearly M is a von Neumann subalgebra of L (G). We shall ﬁrst show that ∞ ¯ ∆G(M) ⊂ L (G) ⊗ M. Let x ∈ M. Then

(id ⊗ id ⊗ ∆G)(id ⊗ ∆G)(∆G(x)) = (∆G ⊗ id ⊗ id)(id ⊗ ∆G)(∆G(x)) = (∆ ⊗ id ⊗ id)(id ⊗ ∆op)(∆ (x)) G G G = (id ⊗ id ⊗ ∆op)(id ⊗ ∆ )(∆ (x)) G G G ∞ ¯ and we get ∆G(x) ∈ L (G) ⊗ M. Using (1.1.3) and Deﬁnition 1.3.3 we see that ∆ | = ∆op| . In particular ∆ (M) ⊂ G M G M G M ⊗¯ M. G G G G The τt -invariance of M follows easily from the relation ∆G ◦ τt = (τt ⊗ τt ) ◦ ∆G. Since ∆ ◦ RG = (RG ⊗ RG) ◦ ∆op and ∆ | = ∆op| we get RG(M) ⊂ M. Summarizing G G G M G M M forms a Baaj-Vaes subalgebra. If N ⊂ L∞( ) is a Baaj-Vaes subalgebra such that ∆ | = ∆op| then it is clear that G G N G N for all x ∈ N the condition (id ⊗ ∆op)(∆ (x)) = (id ⊗ ∆ )(∆ (x)) holds, i.e. N ⊂ M. G G G G ∞ Remark 1.3.4. In other words, M ∈ L (G) introduced in Theorem 1.3.3 is the largest cocommutative Baaj-Vaes subalgebra. This means that it corresponds to the largest classical closed quantum subgroup of Gb . 56 Chapter 1. Preliminaries

Example 1.3.5. Let G be a classical locally compact group. Adopting the notation of 3 Proposition 1.3.3 we see that f ∈ M if for all (up to measure zero subset) (p, q, r) ∈ G we have f(pqr) = f(prq). This condition is equivalent with f(pqrq−1r−1) = f(p). Thus f ∈ M if and only if f is constant on the cosets of the commutator subgroup N ⊂ G where −1 −1 N is defined as the smallest closed subgroup of G containing {qrq r : q, r ∈ G}. In ∞ conclusion, we have M = L (G/N). ∞ Remark 1.3.6. Now, let G be a locally compact quantum group and M ⊂ L (G) the Baaj-Vaes algebra described in Proposition 1.3.3. In general a normal quantum subgroup ∞ N E G such that M = L (G/N) does not exist. Actually such N exists if and only if M, ∞ viewed as a coideal L (G), is normal; the normality of M in turn is equivalent with the equality ∗ ∗ ∗ ∗ G G G G G G G G W13W14(id⊗∆G)(∆G(x))234W14 W13 = W14W13(id⊗∆G)(∆G(x))234W13 W14 (1.3.4) being satisfied for all x ∈ M. Example 1.3.7 below shows that (1.3.4) does not always hold; when it does, we call N E G as above the commutator subgroup of G. Example 1.3.7. As indicated in Remark 1.3.6 above, the largest cocommutative Baaj- ∞ ∞ Vaes subalgebra of L (G) is not, in general, of the form L (G/N) for a normal closed quantum subgroup N E G. To see this, note that upon dualizing, the claim takes the form, that there exist locally compact quantum groups with the property that the largest classical closed quantum subgroup is not normal (see Remark 1.3.4). + For examples of this latter phenomenon, consider one of the free unitary groups Un for some n ≥ 2 (these are the quantum groups whose underlying CQG algebras Au(n) are defined in [106] as being freely generated by n × n unitary matrix of generators uij such ∗ that (uij) is also unitary). It’s largest classical quantum subgroup is the ordinary unitary group Un obtained as the object dual to the largest commutative CQG quotient algebra of Au(n), whereas it is + known [18, Corollary 12] that proper normal quantum subgroups of Un are contained in + the common center T of Un < Un . Let us move on to the discussion of morphisms of locally compact quantum groups. ∗ This requires the universal C -version of a given locally compact quantum group G (see u e.g. [65]). The universal version C0 (G) of C0(G) is equipped with a comultiplication ∆u ∈ Mor(Cu( ), Cu( ) ⊗ Cu( )). The multiplicative unitary WG ∈ M(C ( ) ⊗ C ( )) G 0 G 0 G 0 G 0 Gb 0 G V V G u u admits the universal lift ∈ M(C0 (Gb ) ⊗ C0 (G)). The reducing morphisms for G and Gb will be denoted by u ΛG ∈ Mor(C0 (G), C0(G)), u Λ ∈ Mor(C0 (Gb ), C0(Gb )) Gb respectively. Then (Λ ⊗ Λ )(V V G) = WG Gb G We shall also use the half-lifted versions of WG

WG V V G u = (id ⊗ ΛG)( ) ∈ M(C0 (Gb ) ⊗ C0(G)), W G V V G u = (Λ ⊗ id)( ) ∈ M(C0(Gb ) ⊗ C0 (G)) Gb which satisfy the appropriate versions of the pentagonal equation

WG WG G G WG 12 13W23 = W23 12, 1.3. Lattice of closed quantum subgroups: basic facts 57

G W G W G W G G W12 13 23 = 23W12.

r,u The half-lifted versions of the comultiplications will be denoted by ∆r ∈ Mor(C0(G), C0(G)⊗ u r,u u C0 (G)) and ∆b r ∈ Mor(C0(Gb ), C0(Gb ) ⊗ C0 (Gb )), e.g.

∗ r,u W G W G ∆r (x) = (x ⊗ id) , x ∈ C0(G).

We have

u r,u (ΛG ⊗ id) ◦ ∆G = ∆r ◦ ΛG, u r,u (1.3.5) (Λ ⊗ id) ◦ ∆ = ∆b r ◦ Λ . Gb Gb Gb Given two locally compact quantum groups G and H, a morphism Π: H → G (see e.g. ∗ u u [71]) is represented by a C -morphism π ∈ Mor(C0 (G), C0 (H)) intertwining the respective coproducts:

(π ⊗ π) ◦ ∆G = ∆H ◦ π. It can be equivalently described via:

∞ ∞ • a bicharacter from H to G, i.e. a unitary V ∈ L (Gb ) ⊗¯ L (H) such that

(∆ ⊗ idC ( ))(V ) = V23V13, Gb 0 H

(id ⊗ ∆ )(V ) = V12V13. C0(Gb) H

In fact V ∈ M(C0(Gb ) ⊗ C0(H)) and V = (id ⊗ ΛH ◦ π)( W G). We shall also use V W G u = (id ⊗ π)( ) ∈ M(C0(Gb ) ⊗ C0 (H)). ∞ ∞ ∞ • a right quantum group homomorphism i.e. an action α :L (G) → L (G) ⊗¯ L (H) ∞ of H on L (G) satisfying

(∆G ⊗ id) ◦ α = (id ⊗ α) ◦ ∆G In fact α(x) = V (x ⊗ 1)V ∗. We shall also use the obvious left version of the concept of a right quantum group homomorphism, which is refereed to as a left quantum group homomorphism.

Let Π: H → G. The right quantum group homomorphism assigned to Π will be denoted αΠ or αH→G when convenient. ∞ Example 1.3.8. Let G be a locally compact quantum group and H ≤ G. Since L (Hb ) is ∞ ∞ ∞ a Baaj-Vaes subalgebra of L (Gb ), the multiplicative unitary WH ∈ L (Hb ) ⊗¯ L (H) can ∞ ∞ be viewed as an element V ∈ L (Gb ) ⊗¯ L (H). Since V is a bicharacter from H to G, H ≤ G yields a morphism from H to G. Let G and H be locally compact quantum groups and Π: H → G a morphism. We say that Π identiﬁes H with a closed quantum subgroup of G if there exists a normal injective ∞ ∞ ∗-homomorphism γ :L (Hb ) → L (Gb ) such that V = (γ ⊗ id)(WH) (see [28]).

Clearly, a closed quantum subgroup H ∈ QS(G) is normal if and only if

∞ ∞ ∞ β(L (Hb )) ⊂ L (Hb ) ⊗¯ L (G) where β is the adjoint action (1.1.2). Let us also note the following result whose classical version is well known. 58 Chapter 1. Preliminaries

Proposition 1.3.9. Let G be a locally compact quantum group and N E G an abelian ∞ normal quantum subgroup of G. Then for every x ∈ L (Nb) we have

∗ ∞ ∞ WG(x ⊗ 1)WG ∈ L (Nb) ⊗¯ L (G/N)

∞ In particular the restriction of the adjoint action β to L (Nb) gives rise to the action of ∞ ∞ G/N on L (Nb). Conversely if the adjoint action restricted to L (Nb) gives rise to the action of G/N then N is abelian. ∞ ∞ ∞ Proof. Let α :L (G) → L (G) ⊗¯ L (N) be the right quantum group homomorphism G G N ∞ assigned to N E G. Since (id ⊗ α)(W ) = W12W13 and N is abelian (i.e. L (Nb) is commutative), we conclude that

G 1 G∗ G N 1 1 N∗ G∗ G 1 G∗ 1 (id ⊗ α)(W (x ⊗ )W ) = W12W13(x ⊗ ⊗ )W13 W12 = (W (x ⊗ )W ) ⊗

∞ ∗ ∞ ∞ for all x ∈ L (Nb). Thus WG(x ⊗ 1)WG ∈ L (Nb) ⊗¯ L (G/N). Conversely, the condition

G N 1 1 N∗ G∗ G 1 G∗ 1 W12W13(x ⊗ ⊗ )W13 W12 = (W (x ⊗ )W ) ⊗

∗ ∞ holds if and only if WN(x ⊗ 1)WN = (x ⊗ 1) for all x ∈ L (Nb), which is equivalent to N being abelian.

Let Π: H → G be a morphism of locally compact quantum groups. It turns out that Π cannot (in general) be assigned with a quantum analog of the kernel subgroup ker Π ≤ H 2 (the case H = G and Π being a projection Π = Π was thoroughly studied in [56]). In particular Π cannot be assigned with the exact sequence

• → ker Π → H → H/ ker Π → •. (1.3.6)

As noted in [53], the quantum analog of H/ ker Π can always be constructed. In what follows we shall provide a number of descriptions of H/ ker Π and formulate the condition which yields the existence of ker Π entering the exact sequence (1.3.6). ∞ The von Neumann algebra L (H/ ker Π) is deﬁned as (see [53, Deﬁnition 4.4]

∞ ∞ σ−cls L (H/ ker Π) = {(ω ⊗ id)(V ): ω ∈ L (Gb )∗} . (1.3.7)

∞ To be more precise the right hand side of (1.3.7) forms a Baaj-Vaes subalgebra of L (H), thus yields a locally compact quantum group which we denote H/ ker Π. Since V = (id ⊗ ΛH ◦ π)( W G) the following holds

∞ u σ−cls L (H/ ker Π) = {ΛH(π(x)) : x ∈ C0 (G)} (1.3.8)

∞ which is the second description of L (H/ ker Π). The third description is the subject of [53, Theorem 4.7]:

∞ ∞ ∞ 00 L (H/ ker Π) = {(ω ⊗ id)(α(x)) : ω ∈ L (G)∗, x ∈ L (G)} . (1.3.9)

In what follows we shall give a simple proof of a slightly stronger version of (1.3.9).

Lemma 1.3.10. Given a morphism Π: H → G we have

∞ ∞ ∞ σ−cls L (H/ ker Π) = {(ω ⊗ id)(α(x)) : ω ∈ L (G)∗, x ∈ L (G)} 1.3. Lattice of closed quantum subgroups: basic facts 59

Proof. The bicharacter equation for V yields

G ∗ G V23W12V23 = W12V13.

In particular, since

∞ G ∞ σ−cls L (G) = {(µ ⊗ id)(W ): µ ∈ L (Gb )∗} we have

∞ ∞ σ−cls {(ω ⊗ id)(α(x)) : ω ∈ L (G)∗, x ∈ L (G)} G ∗ ∞ ∞ σ−cls = {(µ ⊗ ω ⊗ id)(V23W12V23): µ ∈ L (Gb )∗, ω ∈ L (G)∗} G ∞ ∞ σ−cls = {(µ ⊗ ω ⊗ id)(W12V13): µ ∈ L (Gb )∗, ω ∈ L (G)∗} ∞ ∞ σ−cls = {(µ ⊗ ω ⊗ id)(V13): µ ∈ L (Gb )∗, ω ∈ L (G)∗} ∞ σ−cls ∞ = {(ω ⊗ id)(V ): ω ∈ L (G)∗, } = L (H/ ker Π) where in third equality σ − cls in the subscript and unitarity of WG enabled us to absorb WG into the functional µ ⊗ ω without changing the resulting set.

∞ ∞ Let Π: H → G. Then the embedding L (H/ ker Π) ⊂ L (H) can be interpreted as H\/ ker Π ≤ Hb . In particular a short exact sequence (1.3.2) starting with

• → H\/ ker Π → Hb

∞ ∞ exists if and only if H\/ ker Π ∈ N QS(Hb ). In this case deﬁning L (ker[ Π) = cd(L (H/ ker Π)) ∈ N QS(H) we get a short exact sequence of locally compact quantum groups (1.3.6). A morphism Π: H → G is assigned with the dual morphism Π:b Gb → Hb which in ∗ terms of bicharacter is given by Vb = σ(V ) . The locally compact quantum group Gb / ker Πb will be denoted by imΠ (see [53, Deﬁnition 4.3]). In particular, using (1.3.7) we can see ∞ ∞ ∞ ∞ that V ∈ L (Gb ) ⊗¯ L (H) is actually an element of L (imdΠ) ⊗¯ L (H/ ker Π). Using the inclusions

∞ ∞ ∞ ∞ L (imdΠ) ⊗¯ L (H/ ker Π) ⊂ L (Gb ) ⊗¯ L (H/ ker Π) ∞ ∞ ∞ ∞ L (imdΠ) ⊗¯ L (H/ ker Π) ⊂ L (imdΠ) ⊗¯ L (H) we see that a morphism Π : → induces three morphisms Π , Π H→G H G H/ ker Π→imΠ H/ ker Π→G and Π . Using [53, Theorem 6.2, Corollary 6.5] we shall now formulate the First Iso- H→imΠ morphism Theorem for locally compact quantum groups.

Theorem 1.3.11. Let H and G be locally compact quantum groups, Π: H → G a morphism and let Π / ker Π→imΠ, ΠH/ ker Π→G, Πb be the morphisms induced by Π as H imdΠ→H described above. Then the following conditions areb equivalent:

(i) Π is an isomorphism; H/ ker Π→imΠ ∞ ∞ ∞ (ii) the action α :L (imΠ) → L (imΠ) ⊗¯ L (H/ ker Π) is integrable;

(iii) ΠH/ ker Π→G identiﬁes H/ ker Π with a closed quantum subgroup of G;

(iv) Πb identiﬁes imdΠ with a closed quantum subgroup of Hb . imdΠ→Hb 60 Chapter 1. Preliminaries

Remark 1.3.12. Let Π: H → G be a morphism of locally compact quantum groups. ∞ Clearly imΠ is abelian if and only if H/ ker Π is abelian. Let M ⊂ L (G) be the Baaj- Vaes algebra described in Proposition 1.3.3. Then H/ ker Π is abelian if and only if ∞ L (H/ ker Π) ⊂ M. For further discussion let us suppose that ker Π E G exists (see ∞ the paragraph containing (1.3.6)) and there exists a N E G such that M = L (G/N). ∞ ∞ Then L (H/ ker Π) ≤ L (G/N) if and only if N ≤ ker Π. Thus in the discussed case we get a quantum analog of the well known classical fact: the closed image of Π: H → G is abelian if and only if the kernel ker Π E H contains the commutator subgroup N E H. The next lemma will be needed further.

Lemma 1.3.13. Let Π: H → G. Then ∞ ∞ 1 cd(L (H/ ker Π)) = {y ∈ L (Hb ): αb(y) = y ⊗ } ∞ ∞ Proof. Let V ∈ L (Gb ) ⊗¯ L (H) be the bicharacter corresponding to Π. The right quan- ∞ ∞ ¯ ∞ tum group homomorphism αb :L (Hb ) → L (Hb ) ⊗ L (Gb ) corresponding to Πb is given by ∗ 1 αb(y) = σ(V ( ⊗ y)V ) ∞ 1 for all y ∈ L (Hb ). In particular αb(y) = y ⊗ if and only if (1 ⊗ y)V = V (1 ⊗ y)

We conclude using (1.3.7).

Example 1.3.14. Let us consider H ∈ N QS(G) and the exact sequence

• → H → G → G/H → •.

∞ Let us denote the morphism G → G/H by Π. Since L (G/H) is defined as a Baaj-Vaes ∞ subalgebra of L (G), the dual morphism Π:b G[/H → Gb identifies G[/H with a closed quantum subgroup of Gb . In particular imΠ = G/H and G/ ker Π = G/H. ∞ ∞ ¯ ∞ Let αb :L (Gb ) → L (Gb ) ⊗ L (G[/H) be the right quantum group homomorphism assigned to Π:b G[/H → Gb . Since Πb identifies G[/H with a closed quantum subgroup of Gb we get

∞ ∞ ∞ σ−cls L (G[/H) = {(ω ⊗ id)(αb(a)) : ω ∈ L (Gb )∗, a ∈ L (Gb )} . (1.3.10) Finally, using Lemma 1.3.13 we see that

∞ ∞ 1 L (Hb ) = {y ∈ L (Gb ): αb(y) = y ⊗ }. (1.3.11)

Lemma 1.3.15. Let G and K be locally compact quantum groups Π1 : G → K a ho- ∞ ∞ ¯ ∞ momorphism and αb1 :L (Gb ) → L (Gb ) ⊗ L (Kb ) the corresponding right quantum group homomorphism. Let H ∈ QS(G), Π2 : H → K the restriction of Π1 to H ≤ G and ∞ ∞ ¯ ∞ αb2 :L (Hb ) → L (Hb ) ⊗ L (Kb ) the right quantum group homomorphism corresponding to Πb 2. Then α2 = α1| . b b L∞(Hb) ∞ ∞ ∞ ∞ Proof. Let V ∈ L (Kb ) ⊗¯ L (G) and U ∈ L (Kb ) ⊗¯ L (H) be the bicharacters corre- V u sponding to Π1 and Π2 respectively. Let ∈ M(C0(Kb ) ⊗ C0 (G)) be the universal lift of V : ∗ W G V W G V12 23V12 = 13 23. (1.3.12) 1.3. Lattice of closed quantum subgroups: basic facts 61

u u Applying quantum group morphism π ∈ Mor(C0 (G), C0 (H)) (corresponding to H ≤ G) to the third leg of (1.3.12) and reducing the result we get

∗ H H ∗ H V12W23V12 = U13W23 = U12W23U12 (1.3.13) ∞ ∞ (note that we use the embedding L (Hb ) ⊂ L (Gb ) on the left side of (1.3.13)). We conclude by recalling that αb1 is implemented by Vb and αb2 is implemented by Ub. We shall also need the following

Lemma 1.3.16. Let G be a locally compact quantum group, N ∈ N CI(G) and M ∈ BV(G). ∞ Let H be a locally compact quantum group such that M = L (H). Then ∗ ∞ WH(1 ⊗ N)WH ⊂ L (Hb ) ⊗¯ N. (1.3.14) ∞ In particular N ∧ L (H) ∈ N CI(H). Moreover ∞ σ−cls N ∨ L (H) = {xy : x ∈ N, y ∈ M} . (1.3.15) ∞ ∞ Proof. As explained in Example 1.3.8 the embedding L (H) ⊂ L (G) corresponds to ∞ a morphism Π: G → H. Let Π:b Hb → Gb be the dual morphism and αb :L (Gb ) → ∞ ¯ ∞ L (Gb ) ⊗ L (Hb ) the corresponding right quantum group homomorphism. Applying (αb ⊗ id) to the normality condition

∗ ∞ WG(1 ⊗ N)WG ⊂ L (Gb ) ⊗¯ N and using G H G (αb ⊗ id)(W ) = W23W13 we get H G 1 1 G∗ H∗ ∞ ∞ W23W13( ⊗ ⊗ N)W13 W23 ∈ L (Gb ) ⊗¯ L (Hb ) ⊗¯ N. (1.3.16) Using (1.1.1) in the context of the Gb -action α : N 3 x → WG(1 ⊗ x)WG∗ we get G G∗ ∞ σ−cls N = {(ω ⊗ id)(W (1 ⊗ x)W ): ω ∈ L (Gb )∗, x ∈ N} . Thus (1.3.16) implies that

∗ ∞ WH(1 ⊗ N)WH ⊂ L (Hb ) ⊗¯ N. ∞ ∗ 2 Let us ﬁx x ∈ N and y ∈ L (H) of the form y = (ωp,q ⊗ id)(WH ) where p, q ∈ L (H). In order to check that (1.3.15) holds it suﬃces to check that

∞ σ−cls xy ∈ {L (H)N} . (1.3.17) Indeed the latter enables us to conclude that the right hand side of (1.3.15) forms a von ∞ Neumann algebra and this suﬃces since N ∨ L (H) is the von Neumann algebra generated ∞ by N and L (H). In the following computation we ﬁx an orthonormal basis (ei)i∈I of 2 L (H) ∗ xy = x(ωp,q ⊗ id)(WH ) ∗ ∗ = (ωp,q ⊗ id)(WH (WH(1 ⊗ x)WH )) P H∗ H 1 H∗ = i∈I (ωp,ei ⊗ id)(W )(ωei,q ⊗ id)(W ( ⊗ x)W ). This computation together with (1.3.14) shows that (1.3.17) indeed holds which ends the proof. 62 Chapter 1. Preliminaries

Deﬁnition 1.3.17. Let G be a locally compact quantum group and H, M ∈ QS(G). We say that H is normalized by M if H ∈ N QS(H ∨ M).

Let H and M be as in Deﬁnition 1.3.17. Using Lemma 1.3.16 we get

∞ ∗ ∞ ∞ WMb (1 ⊗ L (Hb ))WMb ⊂ L (M) ⊗¯ L (Hb ). (1.3.18)

More generally the following holds.

Lemma 1.3.18. Let G be a locally compact quantum group, H, M ∈ QS(G) and suppose that H is normalized by M. Let L ≤ M. Then H is normalized by L. Moreover

∞ ∞ ∞ σ−cls L (H\∨ L) = {xy : x ∈ L (Lb), y ∈ L (Hb )} and H ∧ L ∈ N QS(L). In particular if H ∈ N QS(G) then H is normalized by every L ∈ QS(G) and H ∧ L ∈ N QS(L).

Proposition 1.3.19. Let G be a locally compact quantum group and H, M ∈ QS(G) and ∞ ∞ ¯ ∞ suppose that H is normalized by M. Then αM→G(L (G/H)) ⊂ L (G/H) ⊗ L (M). Proof. Let us ﬁrst recall that

M 1 M∗ αM→G(x) = W (x ⊗ )W

∞ ∞ Let us ﬁx x ∈ L (G/H), i.e. x ∈ L (G) and

WH(x ⊗ 1) = (x ⊗ 1)WH.

We have to prove that

H M 1 1 M∗ M 1 1 M∗ H W13W12(x ⊗ ⊗ )W12 = W12(x ⊗ ⊗ )W12 W13. (1.3.19)

Using (1.3.18) we see that

M∗ H M ∞ ∞ ∞ W12 W13W12 ⊂ L (Hb ) ⊗¯ L (M) ⊗¯ L (H) (1.3.20) and we compute

H M 1 1 M∗ M M∗ H M 1 1 M∗ W13W12(x ⊗ ⊗ )W12 = W12W12 W13W12(x ⊗ ⊗ )W12 M 1 1 M∗ H M M∗ = W12(x ⊗ ⊗ )W12 W13W12W12 M 1 1 M∗ H = W12(x ⊗ ⊗ )W12 W13

∞ 0 where in the second equality we use (1.3.20) and the fact that x ∈ L (Hb ) Remark 1.3.20. Note that in the proof of Proposition 1.3.19 we needed somewhat less than Deﬁnition 1.3.17: it is enough to have

∞ ∗ ∞ ∞ WMb (1 ⊗¯ L (Hb ))WMb ⊂ L (M) ⊗ L (Hb ), which is more akin to the classical notion of one group normalizing another.

The following simple observation regarding a universal property of quotient quantum groups will come in handy repeatedly in Chapter3. 1.3. Lattice of closed quantum subgroups: basic facts 63

Lemma 1.3.21. Let Π: G → P be a morphism of quantum groups, and N ∈ N QS(G). Then, Π factors as Π G P G/N if and only if the composition N → G → P is trivial (i.e the image of the composition is the trivial group).

Proof. The direct implication is clear. Conversely, suppose the composition N → G → P is trivial. We will apply Lemma 1.3.15 to Π1 = Π and H = N ∈ QS(G). The right quantum ∞ ∞ ¯ ∞ group homomorphism αb :L (Gb ) → L (Gb ) ⊗ L (Pb) assigned to Πb is given by ∗ ∗ αb(x) = σ(V )(x ⊗ 1)σ(V ), ∞ ∞ ∞ where x ∈ L (Gb ) and V ∈ L (Pb) ⊗¯ L (G) is the bicharacter assigned Π. Using Lemma ∞ ∞ 0 ∞ 1.3.15 we conclude that V is contained in L (Pb) ⊗¯ (L (Nb) ∩ L (G)), and hence must be ∞ ∞ ∞ ∞ contained in L (Pb) ⊗¯ L (G/N). In particular V viewed as an element L (Pb) ⊗¯ L (G/N) deﬁnes a morphism G/N → P. Running through the way in which bicharacters, regarded as morphisms, compose in the category of locally compact quantum groups, this means precisely that Π factors through G/N. Remark 1.3.22. From the perspective of the category of locally compact quantum groups, Lemma 1.3.21 simply says that G → G/N is the coequalizer of the inclusion N → G and the trivial map N → 1 → G. Moreover, by the self-duality of the category of locally compact quantum groups, we can conclude that the inclusion N → G of a normal subgroup is the equalizer of the arrows G → G/N and G → 1 → G/N. In fact, we can improve on Lemma 1.3.21 somewhat. For future reference, we record the result in Lemma 1.3.23 below. Before its formulation let us consider an action α : N → ∞ N ⊗¯ L (G) of a locally compact quantum group G on a von Neumann algebra N. Then ∞ ∞ ∞ given a left quantum group homomorphism γ :L (G) → L (H) ⊗¯ L (G) there exists a ∞ unique action β : N → N ⊗¯ L (H) such that (id ⊗ γ) ◦ α = (β ⊗ id) ◦ α. ¯ ∞ In particular, given H ≤ G we get β which we denote by α|H : N → N ⊗ L (H) and we say that α|H is the restriction of α to H. The details yielding the existence of β are left to the reader.

∞ Lemma 1.3.23. Let α : N → N ⊗¯ L (G) and N E G a closed normal subgroup. Then, α factors as α ∞ N N ⊗¯ L (G) ∞ N ⊗¯ L (G/N) through an action by G/N on N if and only if N acts trivially on N. Proof. Once again, one implication is trivial, so we prove the other one; that is, we assume ∞ ∞ ¯ ∞ that the restriction α|N of α to N is trivial. Let γ :L (G) → L (N) ⊗ L (G) be the left quantum group homomorphism assigned to N E G. Using the identity (α|N ⊗ id) ◦ α = ¯ ∞ (id ⊗ γ) ◦ α and the trviality of α|N we conclude that α(N) ⊂ N ⊗ L (N\G) where ∞ ∞ L (N\G) = {x ∈ L (G): γ(x) = 1 ⊗ x}. 64 Chapter 1. Preliminaries

∞ ∞ We conclude by noting that normality of N yields L (N\G) = L (G/N) (see e.g. [51, Eq. (1.4)]).

Remark 1.3.24. In phrasing of [53, §4.3], Lemma 1.3.23 says that the quotient of G by the kernel of α factors through G/N if and only if α|N is trivial. In addition to Lemma 1.3.21, Proposition 1.3.9 above is also a consequence of Lemma 1.3.23; in eﬀect, the intuitive content of that proposition is that since the abelian normal subgroup N E G acts trivially on itself by conjugation, the conjugation action of G on N descends to a (G/N)-action. Now, we shall discuss open quantum subgroups. Let us begin with [51, Deﬁnition 2.2].

∞ Definition 1.3.25. Let H and G be locally compact quantum groups and π :L (G) → ∞ L (H) a normal surjective ∗-homomorphism. We say that π identifies H with an open quantum subgroup of G if ∆H ◦ π = (π ⊗ π) ◦ ∆G. Let H be a locally compact quantum group which is identified with an open quantum ∞ ∞ subgroup of G via π :L (G) → L (H). The central support of π (i.e. the smallest ∞ 1 1 projection of P ∈ L (G) such that π(P ) = ) will be denoted by H and it will be refereed to as a group-like projection assigned to π. The morphism π defines a morphism Π: H → G ∞ ∞ which in terms of the bicharacter is given by V = (id ⊗ π)(WG) ∈ L (Gb ) ⊗¯ L (H). Using [51, Theorem 3.6] we see that Π identifies H with a closed quantum subgroup of G as described in Example 1.3.8. Let H ∈ QS(G). Then as proved in [53, Corollary 3.4], H can be identified with an open quantum subgroup of G, (we shall say shortly that H is open in G) if and only if ∞ the Haar weight ψGb restricts to the Haar weight on L (Hb ). In other words if and only ∞ ∞ if the restriction of ψGb to L (Hb ) ⊂ L (Gb ) is semifinite. This in turn is equivalent with ∞ ∞ ∞ the existence of a conditional expectation T :L (Gb ) → L (Gb ) onto L (Hb ) satisfying (see [52, Theorem 6.1] (T ⊗ id) ◦ ∆ = ∆ ◦ T = (id ⊗ T ) ◦ ∆ . (1.3.21) Gb Gb Gb The subset of QS(G) that consists of open quantum subgroups of G will be denoted by OQS(G). In what follows we shall investigate the structure of OQS(G) showing in particular that it forms a lattice.

Proposition 1.3.26. Let G be a locally compact quantum group, H ∈ OQS(G) and M ∈ QS(G). Then H ∧ M ∈ OQS(M). Moreover, if

∞ ∞ 0 ∞ ∞ T :L (Gb ) → L (Hb ) and T :L (Mc) → L (H\∧ M) are the expectations associated to the respective Haar weights, we have

T 0 = T | . L∞(Mb) ∞ ∞ Proof. Let T :L (Gb ) → L (Hb ) be the conditional expectation assigned to H ∈ OQS(G) ∞ ∞ (see (1.3.21)). Consider the restriction of T to L (Mc) ≤ L (Gb ). The equality (see (1.1.3)) ∞ ∞ ∞ σ−cls L (Mc) = {(ω ⊗ id)(∆ (x)) : ω ∈ L (Gb )∗, x ∈ L (Mc)} Gb together with (see (1.3.21))

(id ⊗ T ) ◦ ∆ = (T ⊗ id) ◦ ∆ Gb Gb 1.3. Lattice of closed quantum subgroups: basic facts 65

∞ ∞ ∞ ∞ ∞ imply that T (L (Mc)) ⊂ L (Mc). Thus we get an inclusion T (L (Mc)) ⊂ L (Hb ) ∧ L (Mc). On the other hand, this restriction is the identity on

∞ ∞ ∞ L (Hb ) ∧ L (Mc) = L (H\∧ M).

∞ In conclusion, T | ∞ is an expectation onto L (H\∧ M) which clearly satisﬁes (1.3.21). L (Mb) Both claims of the conclusion now follow from [52, Theorem 6.1].

Let us also note in passing that in the particular case when H ≤ M we obtain:

Corollary 1.3.27. If H ≤ M ≤ G is a sequence of closed quantum group embeddings with H open in G then H ≤ M is also open. In fact, we can improve on Corollary 1.3.27 as in the next result, proving a strong transitivity of openness.

Proposition 1.3.28. Let H ≤ M ≤ G be a chain of closed embeddings of locally compact quantum groups. Then, H is open in G if and only if

H ≤ M and M ≤ G are both open.

Proof. In order to see the leftward implication ‘⇐’ we use [53, Corollary 3.4]. Let us check the rightward implication ‘⇒’. We have already seen in Corollary 1.3.27 that H ≤ M is open. On the other hand, [53, Theorem 3.3] shows that M ≤ G is open if and only if there is some non-zero element of

∞ ∞ L (Mc) ≤ L (Gb ) that is square-integrable with respect to the (either left or right) Haar weight of Gb . An application of the same result in the opposite direction shows that there are such elements in ∞ ∞ ∞ L (Hb ) ≤ L (Mc) ≤ L (Gb ). This concludes the proof.

Using Proposition 1.3.26 and Proposition 1.3.28 we get:

Corollary 1.3.29. The set OQS(G) forms a sublattice of QS(G). u ∗ Remark 1.3.30. Let H ≤ G be an open quantum subgroup and ω ∈ C0 (Gb ) the idempotent state assigned to H ≤ G as described in [52, Remark 6.3]. The group-like projection 1 ∞ H ∈ L (G) assigned to H ≤ G is given by

1 WG H = (ω ⊗ id)( ). Let us denote 1u = (ω ⊗ id)(V )V ∈ M(Cu( )); applying the reducing morphism Λ ∈ H 0 G G Mor(Cu( ), C ( )) we get Λ (1u ) = 1 . Denoting by πu ∈ M(Cu( ), Cu( )) the mor- 0 G 0 G G H H 0 G 0 H phism assigned to H ≤ G we see that u u 1u u u V V (π ⊗ id)(∆ ( H)) = (π ⊗ id)(∆ ((ω ⊗ id)( ))) u = (ω ⊗ id ⊗ id)((id ⊗ π )(V 12V )V 13V ) (1.3.22) 1 1u = ⊗ H 66 Chapter 1. Preliminaries

u where in the third equality we used the identity ω(xπˆ (y)) = ω(x)εb(y)) which holds for all u u x ∈ C0 (Gb ) and y ∈ C0 (Hb ). The latter can be easily concluded from the fact that the image u u u u of the conditional expectation T = (id ⊗ ω) ◦ ∆b is equal to πˆ (C0 (Hb )). Similarly we can prove that (id ⊗ πu)(∆u(1u )) = 1u ⊗ 1. In particular using (1.3.22) H H and (1.3.5) we get

(1 ⊗ 1)∆r,u(1 ) = 1 ⊗ 1u , H r H H H (1.3.23) 1 1 u,r 1 1u 1 ( ⊗ H)∆r ( H) = H ⊗ H. u Now let M ≤ G be a closed quantum subgroup and ρ ∈ Mor(C0 (G), C0(M)) the corresponding morphism. The group-like projection assigned to the open containment 1 ∞ H∧M ≤ M (see Proposition 1.3.26) will be denoted by H∧M ∈ L (M). Using Proposition 1.3.26 we get 1 1u H∧M = ρ( H) (1.3.24) which follows from the computation

1 1 M 0 M W G 1 1u M ( ⊗ H∧M)W = (T ⊗ id)(W ) = (T ⊗ ρ)( ) = ( ⊗ ρ( H))W ∞ ∞ (in this computation we use the embedding L (Mc) ⊂ L (Gb ) and the notation of the proof of Proposition 1.3.26). Denoting the action

α ∈ Mor(C0(G), C0(G) ⊗ C0(M)) describing the embedding M ≤ G we get 1 1 1 1 1 ( H ⊗ )α( H) = H ⊗ H∧M

r,u Indeed, this follows from (1.3.23), (1.3.24) and the the identity α = (id⊗ρ)◦∆r . Similarly we get 1 1 1 1 1 ( ⊗ H)α( H) = H∧M ⊗ H.

1.3.1 Well positioned quantum subgroups

For subgroups H ≤ G, we will be working with the quantum homogeneous spaces (see Remark 1.3.2) ∞ ∞ ∞ AH = L (G/H) = cd(L (Hb )) ⊆ L (G). Deﬁnition 1.3.31. Let H and M be two closed quantum subgroups of a locally compact quantum group G. We say that H and M are (relatively) well positioned if we have the equality σ−cls AH ∨ AM = {AHAM} (1.3.25) (or equivalently its analogue with H and M reversed). As we will see in Theorem 3.3.4, the well positioning property is relevant to the modular law for quantum subgroups of a locally compact quantum group. Here, we discuss suﬃcient conditions that ensure well positioning. Let us also note that in the algebraic context the counterpart of well positioning always holds as noted in Corollary 1.2.10.

Proposition 1.3.32. The closed quantum subgroups H, M ≤ G are relatively well positioned if

(a) G is classical; 1.3. Lattice of closed quantum subgroups: basic facts 67

(b) one of H and M is compact;

(d) G is dual-classical. Proof. We prove the diﬀerent points separately, as they require diﬀerent techniques.

∞ (a) This is immediate: L (G) is then commutative, and hence it does not matter in which order we multiply elements of AH and AM.

(b) The condition is symmetric, so let us assume that H is compact and show that AMAH is linearly dense in AH ∨ AM. We will adapt the proof of Lemma 1.3.16. We write W = WG and ϕ for the Haar state on the compact quantum group H. We further denote by α the canonical coaction

∞ ∞ ∞ L (G) → L (G) ⊗¯ L (H)

∞ ∞ and by V ∈ L (G) ⊗¯ L (H) the bicharacter corresponding to α. Note that the elements of the form

x = (id ⊗ ϕ)α((ωp,q ⊗ id)W) = (ωp,q ⊗ id ⊗ ϕ)(W12V13) (1.3.26)

2 span a dense subset of AH for p, q ∈ L (G). Hence, it suﬃces to show that an element obtained by multiplying x as in (1.3.26) and an arbitrary a ∈ AM belongs to the closure of linear span on AMAH. With this purpose in mind, we ﬁrst use

(id ⊗ α)W = W12V13 again to write

∗ xa = (ωp,q ⊗ id ⊗ ϕ)(W12V13(1 ⊗ a ⊗ 1)) = (ωp,q ⊗ id ⊗ ϕ)(W12(1 ⊗ a ⊗ 1)W12W12V13)). Using X ωpq(•−) = ωp,ei (•)ωei,q(−), i the expression turns into

X ∗ (ωp,ei ⊗ id)(W12(1 ⊗ a)W12) · (ωei,q ⊗ id ⊗ ϕ)(W12V13). i

Now, the left hand side of the ‘·’ symbol belongs to AM by the normality condition ∗ ∞ W(1 ⊗ AM)W ⊆ L (Gb ) ⊗ AM, whereas the right hand side is of the same form as (1.3.26) and hence belongs to AH.

(c) Once again the condition is symmetric, so for the sake of making a choice we ∞ assume H is normal. But then AH is a Baaj-Vaes subalgebra of L (G), and hence the desired result follows from an application of Lemma 1.3.16 (in the form of Lemma 1.3.15) to N = AM and M = AH.

(d) Since G is abelian, quantum subgroups of G are normal and part (c) applies. 68 Chapter 1. Preliminaries

2 Remark 1.3.33. Let G be a locally compact quantum group and N ⊂ B(L (G)) a von ∗ ∞ Neumann algebra such that W(1 ⊗ N)W ⊂ L (Gb ) ⊗¯ N. Let M be a von Neumann subal- ∞ ∞ gebra of L (G) equipped with a conditional expectation E :L (G) → M. Let N ∨ M be the von Neumann algebra generated by N and M. Using the method of the proof of point (b) of Proposition 1.3.32 we get N ∨ M = {NM}σ−cls.

1.4 Preliminaries on ergodic actions and idempotents states of a compact quantum group

∗ In this section, we note C0(X) a C -algebra and X its associated locally compact quantum space. The notion of compact quantum groups has been introduced by Woronowicz. Here we adopt his deﬁnition in [118]. The symbol ⊗ denotes the spatial tensor product of C∗-algebras.

Deﬁnition 1.4.1. A compact quantum group is a pair (A, ∆) where A is a unital C∗- algebra and ∆ : A → A ⊗ A is a unital ∗-homomorphism which is coassociative:

(∆ ⊗ idA) ◦ ∆ = (idA ⊗ ∆) ◦ ∆ and A satisﬁes the quantum cancellation properties:

Lin ((1 ⊗ A) ◦ ∆(A)) = Lin ((A ⊗ 1) ◦ ∆(A)) = A ⊗ A

One of the most important features of compact quantum groups is the existence of a unique Haar state h i.e a unique state h ∈ A∗ such that:

(h ⊗ idA) ◦ ∆(a) = (idA ⊗ h) ◦ ∆(a) = h(a)1 ∀a ∈ A.

A compact quantum groups is said of Kac type if h is tracial i.e h(ab) = h(ba) ∀a, b ∈ A.

Remark 1.4.2. All coidalgebras of a compact quantum group of Kac type are expected i.e there exists a conditional expectation (see Deﬁnition 1.4.8).

∗ A compact quantum group G = (A, ∆) is co-amenable if A is unital, this equivalent to the existence of a non-zero multiplicative functional on A.

Example 1.4.3. O−1(2) (see Deﬁnition 1.4.16) is a coamenable compact quantum group of Kac type.

The following deﬁnition generalising the notion of an idempotent probability measure on a compact group:

Deﬁnition 1.4.4. A state ϕ on a compact quantum group A is said to be an idempotent state if (ϕ ⊗ ϕ) ◦ ∆ = ϕ.

By [39, Theorem 4.1], we have:

Theorem 1.4.5. Let (A, ∆) be a coamenable compact quantum group. Then there exists an order-preserving bijection between the expected right coidalgebras in (A, ∆) and the idempotent states on (A, ∆). 1.4. Preliminaries on ergodic actions and idempotents states of a compact quantum group 69

Remark 1.4.6. The bijection of Theorem 1.5 extends to non-coamenable and non-unimodular locally compact quantum group (See [84, Theorem 1]). In the general case, we must just be careful and consider the idempotent states in the universal version Cu(G) or in P ol(G) (See [38, Theorem 2.3]). On the other hand, the coideals are in the reduced version Cr(G) and there is a one to one correspondence between idempotent states and integrable coideals preserved by the scaling group (See [52, Corollary 4.3]).

Lets now deﬁne the notion of right action, natural conditional expectation, ergodic action, ergodic action of quotient type and embeddable ergodic action in compact quantum group setting. We adopt here the same deﬁnition as K. De Commer in [25].

α Deﬁnition 1.4.7. A (continuous) right action X x G consists of

• a compact quantum group G,

• a locally compact quantum space X, and • a ∗-homomorphism, called right coaction,

α : C0(X) → C0(X) ⊗ C(G)

such that:

– the coaction property holds,

(α ⊗ IdG) ◦ α = (idX ⊗ ∆) ◦ α, and – the following density condition, called Podleś condition, holds,

α (C0(X)) (1X ⊗ C(G)) = C0(X) ⊗ C(G).

α Deﬁnition 1.4.8. Let X x G and Y = X/G. The natural conditional expectation onto C0(Y) is the map ( C0(X) → C0(X) EY : with ϕG the Haar state of the compact quantum group G. a 7→ (idX ⊗ ϕG) α(a)

α Deﬁnition 1.4.9. An action X x G is called ergodic (or homogeneous) if

C(X/G) = {a ∈ C0(X)| α(a) = a ⊗ 1G}C1X.

Remark 1.4.10. Since Y reduces to a point in the case of homogeneous actions, we obtain in particular that the conditional expectation EY becomes a state on C(X). α Deﬁnition 1.4.11. Let X x G. One calls α of quotient type if there exists a compact quantum subgroup H ⊂ G with corresponding quotient map πH : C(G) → C(H) and a ∗-isomorphism

θ : C(X) → C(H \ G) = {g ∈ C(G) | (πH ⊗ idG) ∆(g) = 1H ⊗ g} such that

(θ ⊗ idG) ◦ α = ∆ ◦ θ. 70 Chapter 1. Preliminaries

α Deﬁnition 1.4.12. Let X x G. One calls α embeddable if there exists a faithful ∗- homomorphism θ : C(X) ,→ C(G) such that

(θ ⊗ idG) ◦ α = ∆ ◦ θ. Remark 1.4.13. When seen as subalgebras of C(G), embeddable actions are also referred to as (right) coideal C∗-subalgebras or coidalgebras. By Theorem 1.4.5 and Remark 1.4.2, we deduce that: Lemma 1.4.14. Let (A, ∆) be a coamenable compact quantum group of Kac type. Then there exists an order-preserving bijection between the embeddable ergodic action of (A, ∆) and the idempotent states on (A, ∆).

Lets G be a group and K a ﬁeld. Deﬁnition 1.4.15. A projective representation of G over K is a map π : G → GL(n, K) such that: π(x)π(y) = λ(x, y)π(xy) ∀x, y ∈ G (1.4.1) where λ is called the associated multiplier (or a factor or a 2-cocycle).

In order to study the ergodic actions of O−1(2) (see Chapter4 Section 4.1), let’s ﬁrst recall the deﬁnition of O−1(2).

Deﬁnition 1.4.16. The compact quantum group O−1(2) is deﬁned as the compact matrix quantum group with fundamental copresentation y = (yjk)1≤j,k≤2 and the relations

1. y is orthogonal, i.e. the generators yjk are self-adjoint and satisfy the unitarity relations y1jy1k + y2jy2k = δjk = yj1yk1 + yj2yk2 for j, k = 1, 2;

2. yjkyj` = −yj`yjk and ykjy`j = −y`jykj for k 6= `;

3. yjky`m = y`myjk for j 6= ` and k 6= m.

The coproduct, counit and antipode of O−1(2) are given by: X ∆(yjk) = yji ⊗ yik, ε(yjk) = δjk,S(yjk) = ykj. i The following notion of monoidal equivalence was introduced in [8]. We adopt here the deﬁnition of [30].

Deﬁnition 1.4.17. Two compact quantum groups G1 = (A1, ∆1) and G2 = (A2, ∆2) are said to be monoidally equivalent if there exists a bijection ψ : Irred(G1) → Irred(G2) satisfying ψ(ε) = ε, together with linear isomorphisms

ψ : Mor(x1 ⊗ · · · ⊗ xr, y1 ⊗ · · · ⊗ yk) → Mor(ψ(x1) ⊗ · · · ⊗ ψ(xr), ψ(y1) ⊗ · · · ⊗ ψ(yk)) satisfying the following conditions:

∗ ∗ ψ(1) = 1, ψ(S ) = (ψ(S)) , ψ(ST ) = ψ(S)ψ(T ), ψ(S⊗T ) = ψ(S)⊗ψ(T ) ∀S ⊂ A1 and T ⊂ A2. whenever the formulas make sense. Such a collection of maps ψ is called a monoidal equivalence between G1 and G2. 1.4. Preliminaries on ergodic actions and idempotents states of a compact quantum group 71

Remark 1.4.18. Concrete examples of monoidally equivalent compact quantum groups are given in section 4 of [30].

Lets recall now the principal result of [30]. In fact, in their paper, the autors proved that there is bijective correspondence between ergodic actions of two monoidally equivalent compact quantum group. Let’s recall now [30, Theorem 7.3].

Theorem 1.4.19. Let G1 and G2 be two monoidally equivalent compact quantum group. Then there is a natural bijective correspondence between their ergodic actions.

We ﬁnish this subsection by stating a question motivated by Theorem 1.4.19, and fundamentally related to Section 4.2 and Section 4.3.

Question 1.4.1. Let G1 and G2 be two monoidally equivalent compact quantum group. Do we have a bijective correspondence between their embeddable ergodic actions?

The answer of this question will be the aim of Chapter4 Section 4.4.

Chapter 2

Lattices of quantum subgroups of a linearly reductive quantum group

In this chapter we tackle some analogues of the group isomorphism theorems in the setting of (mostly linearly reductive) linear algebraic quantum groups.

2.1 The second isomorphism theorem

We will prove a version of the second isomorphism theorem [81, Theorem 2.26] for function algebras of linearly reductive quantum groups, i.e. cosemisimple Hopf algebras (see Definition 1.2.1). The general setup is as follows. Recall from Section 1.2 that unless specified otherwise, we work over an algebraically closed field k of arbitrary characteristic. O(G) denotes a cosemisimple Hopf algebra, to be thought of as the algebra of regular functions on a quantum group G. We fit the latter into an exact sequence 1 → K → G → G/K → 1 in the sense that we have an exact sequence

k → O(G/K) → O(G) → O(K) → k as in Definition 1.2.12. Note that O(G/K) is automatically cosemisimple (being a Hopf subalgebra of a cosemisimple Hopf algebra), and hence ([20, Theorem 2.1]) the inclusion O(G/K) → O(G) is automatically faithfully flat both on the left and the right. It then follows [20, Theorem 2.5] that O(K) is itself cosemisimple. Assume now that we have another linearly reductive quantum subgroup H ≤ G, i.e. a quotient cosemisimple Hopf algebra O(G) → O(H). We will now examine the issue of whether or not the intersection H ∧ K from Definition 1.2.5 is linearly reductive. First, define the Hopf subalgebra • of O(H) so as to make the following diagram commute. O(G) (2.1.1) O(G/K) O(H) • 74 Chapter 2. Introduction to linearly reductive quantum group

• is then a Hopf subalgebra of O(H), and hence automatically cosemisimple. It is also invariant under the adjoint actions of O(H) on itself, since O(G/K) is ad-invariant in O(G). This means that • is of the form

O(H/N) ⊆ O(H) for some normal linearly reductive quantum subgroup O(H) → O(N) of H. By construction, we have a morphism H/N → G/K. Our goal is to argue that we have H ∧ K = N, for the intersection operation ∧ as in Deﬁnition 1.2.5. According to Remark 1.2.6, this is achieved by the following result.

Proposition 2.1.1. In the setting above, the diagram

O(H) O(N) O(G) (2.1.2) O(K) is a pushout in the category of algebras, or equivalently, that of bialgebras, or Hopf algebras, or Hopf algebras with bijective antipode.

Proof. For the fact that the forgetful functor from Hopf algebras to bialgebras or algebras is a left adjoint and hence preserves colimits (such as pushouts) we refer to [79]. Hence, we will focus on showing that the diagram is a pushout of algebras. The exactness of the sequence

k → O(G/K) → O(G) → O(K) → k implies that the kernel of the surjection O(G) → O(K) is the ideal

− − O(G)O(G/K) = O(G/K) O(G).

But this means that the pushout of the two right hand arrows of (2.1.2) is the quotient of O(H) by the ideal generated by the kernel of the counit of

O(H/N) := Im(O(G/K) → O(H)).

Finally, the general theory of exact sequences of Hopf algebras as covered in [3] and recalled in Section 1.2 above says that this is precisely right hand quotient in the sequence

k → O(H/N) → O(H) → → k, which is by deﬁnition our O(H) → O(N). Remark 2.1.2. The substance of Proposition 2.1.1 is that the algebra colimit in question is automatically cosemisimple as a Hopf algebra. This is analogous to the classical fact that a normal subgroup of a linearly reductive linear algebraic group is automatically linearly reductive, as follows easily, for instance, from the classiﬁcation of linearly reductive groups [75]. 2.1. The second isomorphism theorem 75

Finally, suppose H and K generate G in the following representation-theoretic sense: O( ) a linear map f : V → W between comodules V,W ∈ M G is a G-intertwiner if and only if it is both an H- and a K-intertwiner (see e.g. [11] for the identical notion of topological generation for compact quantum groups, or [21], where the same property is phrased in terms of the injectivity of the map from O(G) into the product O(K) × O(H) in the category of Hopf algebras).

Remark 2.1.3. One can show that the condition above is equivalent to H ∨ K = G, for the operation ‘∨’ from Deﬁnition 1.2.5.

With all of this in place, we have

Theorem 2.1.4. Let H ≤ G and K E G be linearly reductive quantum subgroups of a linearly reductive quantum group. If H and K generate G, then the canonical morphism H/H ∧ K → G/K is an isomorphism. Proof. By construction, O(G/K) → O(H/H ∧ K) is onto. In order to prove injectivity and complete the proof, it suﬃces to show that the functor from (G/K)-representations to H-representations induced by the the upper composition in (2.1.1) is full, i.e. it induces a bijection between sets of morphisms. The fact that this condition is equivalent to the bijectivity of a map of coalgebras that is known to be onto follows e.g. from [86, Lemmas 2.2.12, 2.2.13]. Let V and W be ﬁnite-dimensional (G/K)-representations, and f : V → W an H- O( / ) intertwiner between them. Since the category M G K of comodules over O(G/K) is the full subcategory of MG consisting of objects that break up as copies of the trivial representation when restricted to K, f is also a K-intertwiner. But then, by the hypothesis that H and K generate G, f is a G- and hence a (G/K)-intertwiner. This completes the proof.

Remark 2.1.5. When working over C and all Hopf algebras in sight are CQG, The- orem 2.1.4 gives an alternate proof of the case of Proposition 3.1.1 when all quantum groups are compact.

We also have a version of Theorem 2.1.4 taking place in DQG.

Theorem 2.1.6. If the subgroup H and the normal subgroup K generate the discrete quantum group G, then the canonical morphism H/H ∧ K → G/K is an isomorphism.

Proof. The hypothesis that H∨K = G means, in the context of algebraic discrete quantum groups, that we have kG = kHkK, and the surjectivity of the canonical map

− − k(H/H ∧ K) = kH/(kH(kH ∩ kK) ) → kG/kGkK = k(G/K) follows from this. As for injectivity, it amounts to showing that those simple kH-comodules that become trivial (i.e. break up as direct sums of copies of the trivial comodule) over k(G/K) are precisely those corresponding to subcoalgebras of kH∩kK; this is immediate, using the fact that a kG-comodule becomes trivial over k(G/K) if and only if it is a kK-comodule. 76 Chapter 2. Introduction to linearly reductive quantum group

Note incidentally that a trivial version of the ﬁrst isomorphism theorem is implicit in the proof of Theorem 2.1.4. For a morphism Π: H → G of locally compact quantum groups, H/ ker Π is essentially the smallest “quotient LCQG” H →? for which Π factors as

Π H G (2.1.3) ?

(see e.g. Deﬁnition 1.3.8) Similarly, imΠ is the smallest ? ≤ G such that Π factors similarly to (2.1.3) as Π H G (2.1.4) ?

2.2 The modular law and Zassenhaus lemma

Throughout this section G denotes a linearly reductive quantum group. We will be interested in studying its poset of subgroups O(G) → O(H). First, recall the modular law for subgroups of a discrete group G: whenever M, N and H are subgroups of G with N ≤ H we have

H ∩ MN = (H ∩ M)N, where the juxtaposition AB of subgroups A, B ≤ G means the set

{ab | a ∈ A, b ∈ B}.

We will be interested in cases where the set products in question are actually subgroups. To this end, we ﬁrst prove

Proposition 2.2.1. Let N ≤ H and M be normal subgroups of the linearly reductive quantum group G. Then, we have

H ∧ (M ∨ N) = (H ∧ M) ∨ N. (2.2.1)

Proof. We will dualize the picture, and study quotient Hopf algebras O(G) → O(•) corresponding to normal quantum subgroups from the perspective of the corresponding Hopf subalgebras A• = O(•\G) ⊆ O(G). This dualization procedure reverses the lattice operations on quotient Hopf algebras and Hopf subalgebras. For this reason, the Hopf subalgebra

A = AH∧(M∨N) corresponding to the left hand side of (2.2.1) is equal to

AH(AM ∧ AN). (2.2.2) Similarly, the Hopf subalgebra

B = A(H∧M)∨N 2.2. The modular law and Zassenhaus lemma 77

corresponding to the right hand side is

(AHAM) ∧ AN. (2.2.3)

Now note that (2.2.2) is the sum of those simple subcoalgebras of O(G) whose simple comodules V ≤ W ⊗ X, where W is a simple AH-comodule and X is a simple comodule over both AM and AN (see the discussion on cosemisimple coalgebras in Section 1.2). On the other hand, the simple comodules of (2.2.3) are characterized by the fact that they are AN-comodules, and also embed into tensor products of the form W ⊗X, for simple comodules A A W ∈ M H ,X ∈ M M . (2.2.4) Clearly, the latter property for a simple comodule V ∈ MO(G) is weaker than the former, and hence A ≤ B. A On the other hand, suppose the simple comodule V ∈ M N embeds into W ⊗ X with W and X as in (2.2.4) (and hence V ∈ MB). Then we have a non-zero morphism

V → W ⊗ X, which by duality gives a non-zero morphism

W ∗ ⊗ V → X

(automatically an epimorphism, since X is assumed to be simple). But since

A A A V ∈ M N ,W ∈ M H ⊆ M N ,

A we get X ∈ M N , and hence X is actually a comodule over

AM ∧ AN. This means that V is actually an A-comodule, and the proof is complete.

Remark 2.2.2. Alternatively, we can restate Proposition 2.2.1 as saying that the normal quantum subgroups of a linearly reductive quantum group form a modular lattice.

We can prove somewhat more when G is a compact quantum group. As noted above, the identity H ∩ MN = (H ∩ M)N holds for all subgroups N ≤ H ≤ G and M ≤ G. Our version (Proposition 2.2.4 below) will still not be as general as this, but we will impose just enough restrictions to ensure that classically, the product sets MN and (H ∩ M)N are actually subgroups. To this end, we need

Deﬁnition 2.2.3. A quantum subgroup L ≤ G normalizes another M ≤ G if the latter is normal in the quantum subgroup M ∨ L. We are now ready to state

Proposition 2.2.4. Let G be a compact quantum group, with quantum subgroups L ≤ H ≤ G and M ≤ G such that L normalizes M. Then, the equality H ∧ (M ∨ L) = (H ∧ M) ∨ L holds. 78 Chapter 2. Introduction to linearly reductive quantum group

Proof. As in the proof of Proposition 2.2.1, the goal is to show that we have

AH(AM ∧ AL) = AHAM ∧ AL, (2.2.5) or rather that the right hand side is contained in the left hand side (the opposite inclusion being immediate). Note that we have used Lemma 1.2.9 and Corollary 1.2.10 implicitly in order to conclude that the subspace products in (2.2.5) are both coideal subalgebras.

For any quantum subgroup π : O(G) → O(K) we have an expectation EK = O(G) → AK deﬁned as

π⊗id O(G) ⊗ O(G) O(K) ⊗ O(G) ∆ hK⊗id

O(G) O(G)

E K AK

It is automatically an AK-bimodule map, and intertwines hG and its restriction to AK. Now consider the expectation EL : O(G) → AL. Applied to an element x in the right hand side of (2.2.5), it ﬁxes x that element (because EL acts as the identity on AL). On the other hand, writing x = xHxM for

xH ∈ H, xM ∈ M, we have

EL(x) = xHEL(xM) because xH ∈ AH ≤ AL and EL is the identity on AL. In conclusion, we would be done if we could show that EL(xM) ∈ AM ∧ AL. This is taken care of by Lemma 2.2.5 below. For the next result we will use the same notation as in the proof of Proposition 2.2.4 for coideal subalgebras AL = O(L\G), expectations EL : O(G) → AL, etc. We will also denote by πL the surjection O(G) → O(L) onto the function algebra of a quantum subgroup. Lemma 2.2.5. Let M and L be quantum subgroups of a compact quantum group G such that L normalizes M. Then, we have

EL(AM) ⊆ AM ∧ AL.

Proof. Since the range of EL is AL, we are trying to show that AM is invariant under EL. To this end, let f ∈ AM be an arbitrary element. This means by deﬁnition that

πM(f1) ⊗ f2 = 1 ⊗ f. (2.2.6) Also by deﬁnition, the expression for the expectation is

EL(f) = hL(πL(f1))f2, and hence our goal is to prove that we have

hL(πL(f1))πM(f2) ⊗ f3 = 1 ⊗ hL(πL(f1))f2. More generally, we will show that in fact we have

πL(f1) ⊗ πM(f2) ⊗ f3 = πL(f1) ⊗ 1 ⊗ f2. (2.2.7) 2.2. The modular law and Zassenhaus lemma 79

Moreover, by substituting L ∨ M for L, we may as well assume that M E L. Using the deﬁning property of the antipode, the left hand side of (2.2.7) equals

πL(f1S(f3)f4) ⊗ πM(f2) ⊗ f5. (2.2.8)

The normality assumption M E L implies by Lemma 1.2.11 that the surjection O(L) → O(M) is one of left O(L)-comodules under the left adjoint coaction

x 7→ x1S(x3) ⊗ x2.

For this reason, (2.2.8) is the result of ﬁrst applying the left adjoint O(L)-coaction to the left hand leg of

πM(f1) ⊗ f2, and then subjecting the result to the operation

• ⊗ • ⊗ 7→ •(πL1) ⊗ • ⊗ 2. (2.2.9) The conclusion now follows from (2.2.6), which ensures that the input of (2.2.9) is 1 ⊗ 1 ⊗ f.

Remark 2.2.6. The proof of Lemma 2.2.5 is a quantum version of the following classical argument that will be much more transparent: ∞ Given a continuous function f on G, the expression for its expectation EL : L (G) → A is L Z (ELf)(Lg) = f(lg) dl. L

We want to argue that if f is invariant under left translation by M, then so is ELf. In order to see this, let m ∈ M. We then have Z Z −1 (ELf)(m−) = f(lm−) dl = f(lml · l−) dl, L L −1 which, because lml ∈ M and f is M-invariant, equals Z f(l−) dl = ELf. L Remark 2.2.7. We note that Proposition 2.2.4 would be problematic in the more general setting of linearly reductive quantum groups (which is why we only have Proposition 2.2.1 in the latter case). The reason is that even classically, intersections of linearly reductive subgroups (such as (H ∧ M) ∨ L) need not be linearly reductive again, as Example 2.2.8 below shows. Example 2.2.8. Let G = SL3 = SL3(C).

Using the correspondence between complex Lie subalgebras of g = sl3 and complex linear algebraic subgroups of G ([72, discussion preceding 3.42 and Theorem 4.22]) and the fact that this correspondence is compatible with intersections ([72, Proposition 3.19] or [9, 6.12]), it suﬃces to exhibit two semisimple Lie subalgebras a and b of g whose intersection is not semisimple. We take the span of

0 0 0 1 0 0  0 0 1       e = 0 0 0 , h = 0 0 0  , f = 0 0 0 (2.2.10) 1 0 0 0 0 −1 0 0 0 80 Chapter 2. Introduction to linearly reductive quantum group

for a. The three displayed elements are a so-called sl2 triple; this means that their identi- ﬁcation with ! ! ! 0 1 1 0 0 0 , , 0 0 0 −1 1 0 ∼ respectively implements an isomorphism a = sl2. Similarly, we take the conjugate of a by

1 1 0   0 1 1 . (2.2.11) 0 0 1 for b. Since (2.2.11) commutes with the leftmost element e of (2.2.10) and conjugates the semisimple element h outside of a (as can easily be seen), the intersection a ∩ b is the one-dimensional span of e, and hence at the level of groups the corresponding intersection is a copy of the (non-linearly reductive) additive algebraic group Ga over C (i.e. just C with its usual additive group structure). There is also a dual version to Proposition 2.2.4, dealing with discrete quantum groups in the sense of Deﬁnition 1.2.1. Quantum subgroups M ≤ G then correspond to Hopf subalgebras

kM ⊆ kG.

Deﬁnition 2.2.9. Let G be an algebraic discrete quantum group. A quantum subgroup L normalizes another M ≤ G if the group algebra kM of the latter is invariant under the adjoint action of kL on kG. Proposition 2.2.10. Let G be an algebraic discrete quantum group, with quantum subgroups L ≤ H ≤ G and M ≤ G such that L normalizes M. Then, the equality (2.2.1) holds. Proof. The proof is essentially the same as that of Proposition 2.2.1, once we substitute kG for O(G) in that result, and similarly substitute the Hopf subalgebras

kH, kL, kM ⊆ kG of kG for AL,AH,AM ⊆ O(G) respectively.

Propositions 2.2.4, 2.2.10 will allow us to prove the following version of the Zassenhaus (or butterﬂy) lemma ([62, Vol. 1, p. 77] or [81, Chapter 5, Lemma 5.10]) for compact and discrete quantum groups.

0 0 Proposition 2.2.11. Let A E A and B E B be quantum subgroups of either a compact or an algebraic discrete quantum group G. Then, we have an isomorphism 0 ∨ ( ∧ ) 0 ∨ ( ∧ ) A A B =∼ B A B . A0 ∨ (A ∧ B0) B0 ∨ (A0 ∧ B) 0 The analogous statement holds for linearly reductive G provided A, A , etc. are all normal in G. 2.3. The Schreier refinement theorem 81

Proof. We focus ﬁrst on the compact / discrete case, following the usual strategy (as in [62, Vol. 1, p. 77] or the proof of [81, Chapter 5, Lemma 5.10], for instance) of proving that we have isomorphisms

0 0 A ∨(A∧B) B ∨(A∧B) A0∨(A∧B0) B0∨(A0∧B) A∧B ∼ 0 0 ∼ = (A ∧B)∨(A∧B ) = By symmetry, it suﬃces to focus on the left hand side of this diagram. The required isomorphism will follow from the compact / discrete quantum version of the second isomorphism theorem (Theorems 2.1.4 and 2.1.6) applied to 0 0 H = A ∧ B and K = A ∨ (A ∧ B ) once we prove that we have 0 0 0 0 (A ∧ B) ∧ (A ∨ (A ∧ B )) = (A ∧ B) ∨ (A ∧ B ). In turn, this follows from Proposition 2.2.4 or Proposition 2.2.10 applied to H = A ∧ B, 0 0 L = A ∧ B and M = A . As for the last claim regarding the linearly reductive case, its proof is virtually identical, using Proposition 2.2.1 instead of Proposition 2.2.4.

2.3 The Schreier reﬁnement theorem

In this section we prove an analogue of the Schreier refinement theorem for compact and discrete quantum groups (see e.g. [81, Chapter 5, Theorem 5.11] for the classical analogue for ordinary discrete groups). To this aim, we need to define a quantum analogue of the notion of (sub)normal series. Definition 2.3.1. Let G be either a compact or (algebraic) discrete quantum group. A finite system G = G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · ≥ Gk = 1 (2.3.1) of closed quantum subgroups of G is called a subnormal series of G if every subgroup Gi is a proper normal closed quantum subgroup of Gi−1 , i ∈ {1, 2, ··· , k}. In particular, G1 is a normal closed quantum subgroup of G, G2 is a normal closed quantum subgroup of G1, but not necessarily of G, and so on. A subnormal series is normal if each Gi is normal in the ambient group G. The corresponding subquotient quantum groups

G1\G, G2\G1, ··· , Gk\Gk−1 of G are the factors of the (sub)normal series (2.3.1). The integer k is the length of the series (2.3.1). Deﬁnition 2.3.2. A subnormal series

G = H0 ≥ H1 ≥ H2 ≥ H3 ≥ · · · ≥ Hl = 1 (2.3.2) is called a refinement of the subnormal series (2.3.1) if every quantum subgroup Gi of (2.3.1) coincides with one of the quantum subgroups Hj, i.e. if every quantum subgroup that occurs in (2.3.1) also occurs in (2.3.2). In particular, every normal series is a refinement of itself. The lengths of the normal series (2.3.1) and its refinement (2.3.2) of course satisfy the inequality k ≤ l. Two subnormal series of a compact quantum groups are called equivalent if their lengths are equal and their constituent subquotients are isomorphic up to permutation. 82 Chapter 2. Introduction to linearly reductive quantum group

We are now ready for the following analogue of Schreier’s refinement theorem. As we will see, its proof, given the Zassenhaus lemma (Proposition 2.2.11) is virtually automatic. Theorem 2.3.3. Any two subnormal series of a compact / discrete quantum group G have equivalent refinements. The same holds for any two normal series of a linearly reductive quantum group. Proof. We focus first on the claim relating to compact and algebraic discrete quantum groups. Let (2.3.1) and (2.3.2) be two normal series of a compact quantum group G, and set

Gij = Gi ∨ (Gi−1 ∧ Hj), Hij = Hj ∨ (Hj−1 ∧ Gi). For i ∈ {1, 2, ··· , k} and j ∈ {1, 2, ··· , l} we obtain two new refinements of (2.3.1) and (2.3.2) respectively: Gi−1 = Gi0 > Gi(j−1) > Gij > Gii = Gi; (2.3.3) Hj−1 = H0j > H(i−1)j > Hij > Gkj = Hj. (2.3.4) By Proposition 2.2.11, Gij is a normal closed quantum subgroup of Gi(j−1) and Hij is normal in H(i−1)j, and moreover ∼ Gij\Gi(j−1) = Hij\H(i−1)j. (2.3.5) The refinements induced by (2.3.3) and (2.3.4) have the same length, and (2.3.5) says that they are equivalent. The proof of the second claim follows similarly, using the corresponding second half of Proposition 2.2.11. Remark 2.3.4. This is simply an adaptation to the quantum setting of the usual proof of the Schreier refinement theorem (see e.g. the proof of [81, Chapter 5, Theorem 5.11]). As mentioned above, once we have the Zassenhaus lemma the standard argument goes through mechanically. The same goes for the Jordan-Hölder theorem below.

2.4 The Jordan-Hölder theorem

In this section we prove analogues of the Jordan-Hölder theorem for compact and discrete quantum groups (and a weaker form of it in the linearly reductive case). We begin with the following deﬁnition.

Definition 2.4.1. A subnormal series (2.3.1) is a composition series of G if Gi is a proper maximal normal closed quantum subgroup of Gi−1 for 1 ≤ i ≤ k. Remark 2.4.2. In other words, a composition series is a subnormal series that cannot be refined further. The main result of this section is Theorem 2.4.3. Any two composition series of a compact or discrete quantum group G are equivalent. Proof. This is immediate from Theorem 2.3.3 together with the observation (made in Remark 2.4.2) that composition series cannot be refined strictly: two composition series have equivalent refinements, and hence they must already be equivalent. Analogously, making use of the second half of Theorem 2.3.3, we have Proposition 2.4.4. Any two normal series of a linearly reductive quantum group which are maximal with respect to refinement are equivalent. Chapter 3

Isomorphism theorems, modular law: the locally compact case

3.1 The second isomorphism theorem

We shall ﬁrst consider the setting of the second isomorphism theorem for ordinary discrete groups, transported to the present framework: G is a locally compact quantum group, H ∈ QS(G) and N ∈ N QS(G). In order to make sense of the statement of Proposition 3.1.1 below, note ﬁrst that according to Lemma 1.3.21, H → G → G/N factors through a morphism

H/H ∧ N → G/N inducing an action of H/H ∧ N on G/N. Proposition 3.1.1. Let G be a locally compact quantum group, H ∈ QS(G) and N ∈ N QS(G). Let us denote by Π: H → G/N the induced morphism. Then H ∧ N = ker Π. If moreover G = H ∨ N then imΠ = G/N.

Proof. Let us consider homomorphisms Π1 : G → G/N and Π: H → G → G/N. The right ∞ quantum group homomorphism assigned to Πb 1 and Πb will be denoted by αb1 :L (Gb ) → ∞ ¯ ∞ ∞ ∞ ¯ ∞ ∞ L (Gb ) ⊗ L (G[/N) and αb :L (Hb ) → L (Hb ) ⊗ L (G[/N) respectively. Viewing L (Hb ) ∞ ∞ ∞ ¯ ∞ as a subalgebra of L (Gb ) we have α :L (Hb ) → L (Hb ) ⊗ L (G[/N) where α = α1| ∞ b b b L (Hb) ∞ ∞ 1 (see Lemma 1.3.15). Let x ∈ cd(L (H/ ker Π)). Then x ∈ L (Hb ) and αb1(x) = x ⊗ . ∞ ∞ ∞ ∞ By Lemma 1.3.13 we have x ∈ L (Nb) ∧ L (Hb ). Thus L (H/ ker Π) = L (H/H ∧ N), i.e. ker Π = H ∧ N. Since G = H ∨ N and N ∈ N QS(G) we conclude from Lemma 1.3.18 that ∞ ∞ ∞ σ−cls L (Gb ) = {L (Nb)L (Hb )} . (3.1.1)

∞ ∞ ∞ Using (1.3.11) we can see that for all ω ∈ L (Gb )∗, x ∈ L (Hb ) and y ∈ L (Nb) we have

(ω ⊗ id)(αb1(xy)) = (y · ω ⊗ id)(αb(x)). (3.1.2) Using (1.3.10), (3.1.1), (3.1.2) we get

∞ ∞ ∞ σ−cls L (G[/N) = {(ω ⊗ id)(αb(a)) : ω ∈ L (Hb )∗, a ∈ L (Hb )} i.e. the closure of the image of Π is also G/N. Chapter84 3. Isomorphism theorems, modular law: the locally compact case

Using Theorem 1.3.11 and Proposition 3.1.1 we get

Corollary 3.1.2. The homomorphism H/H ∧ N → G/N is an isomorphism if and only if ∞ the corresponding action of H/H ∧ N on L (G/N) is integrable.

Remark 3.1.3. The assumptions of Corollary 3.1.2 hold trivially when H is compact. ∞ Assume on the other hand that N is compact. Then, since the action of H on L (G) is ∞ integrable and we have a conditional expectation onto L (G/N), the integrability assumption also holds in this case. Finally, note that every morphism Π of discrete quantum groups automatically identi- ﬁes H/kerΠ with imΠ. In conclusion, Corollary 3.1.2 also goes through when all quantum groups in sight are discrete.

Let us also note that equivalent statements in Corollary 3.1.2 fails (and hence so does the second isomorphism theorem) in general even classically, for locally compact abelian groups, as the following example shows.

2 Example 3.1.4. Consider the group G = T × R, and the subgroups

itθ isφ H = {(e , 1, t) | t ∈ R} and N = {(1, e , s) | s ∈ R} for real numbers θ and φ that are incommensurable (i.e. linearly independent over Q). Then, the subgroup itθ −itφ 2 {(e , e , 0)} ⊂ T × {0} 2 2 of HN is dense T × {0} and hence the closure H ∨ N of HN contains T × {0}. But the product of this latter group with H is clearly all of G, and we have H ∨ N = G. Now, H/H ∧ N is a one-dimensional Lie group whereas G/N is a two-dimensional one, and hence the conditions of Corollary 3.1.2 cannot possibly hold.

The fundamental characteristic of Example 3.1.4 is that the naive product HN is not closed in G, and hence H ∨ N is “larger than expected”. Indeed, classically, it is this failure of HN to be closed that prevents the conditions of Corollary 3.1.2 from holding. This is summarized in the following result.

Proposition 3.1.5. Let G be a classical locally compact group, and H ≤ G and N E G closed subgroups. Then, H/H∧N acts integrably on G/N if and only if for every (H∧N)-invariant closed subset F of H the product FN is closed. Proof. According to Corollary 3.1.2, the integrability of the action in the statement is equivalent to the canonical map

H/(H ∧ N) → (H ∨ N)/N (3.1.3) being an isomorphism. We will use this equivalence throughout the proof, implicitly or explicitly.

(⇐) Suppose FN is closed in G for every closed F ⊆ H. Applying this to F = H ﬁrst, we have H ∨ N = HN and hence the canonical one-to-one morphism (3.1.3) is also onto. Now note that the composition

H → G → G/N 3.1. The second isomorphism theorem 85

realizes H/(H ∧ N) as a closed subgroup of the right hand side. Indeed, it induces an embedding of the former group into G/N, and the condition on FN being closed means precisely that the induced one-to-one map

H/(H ∧ N) → G/N is closed. All in all, (3.1.3) is a bijective inclusion of one closed subgroup of G/N, namely H/(H∧ N), into another, i.e. HN/N. It is then an isomorphism, and the conclusion follows from Corollary 3.1.2.

(⇒) Conversely, suppose the action in question is integrable, and hence by Corollary 3.1.2 the morphism (3.1.3) is bijective. The diagram

H ∨ N H (H ∨ N)/N H/(H ∧ N) =∼ shows that H ∨ N is generated as a (plain, not topological) group by H and N. Since N is normal, this in turn implies H ∨ N = HN, so that the latter product must be closed. Moreover, the fact that (3.1.3) is a homeomorphism implies that it is in particular closed. This means that the image of every closed subset F ⊆ H as in the statement is closed in G/N, and hence its preimage FN through the quotient map G → G/N is closed.

Although quite explicit, the closure condition in Proposition 3.1.5 might be somewhat inconvenient to check. In view of this, one might wonder whether the seemingly weaker condition that HN be closed in G is suﬃcient. Example 3.1.6 shows that this is not the case, even in the case of classical abelian locally compact groups. Before spelling out the example, let us clarify what it is meant to do. Placing ourselves entirely within the context of locally compact abelian groups, consider for simplicity the case when H∧N is trivial. Moreover, we may further assume harmlessly that the subgroup HN ≤ G (which is supposed to be closed anyway) is all of G. All in all, we will have an algebraic decomposition

G = H ⊕ N. (3.1.4) Then, the condition from Proposition 3.1.5 and its symmetric counterpart (i.e. with the roles of H and N interchanged) jointly mean precisely that the decomposition (3.1.4) is one of topological abelian groups as well as abstract ones. In conclusion, in order to show that the closedness of HN does not entail the second isomorphism theorem, it suﬃces to exhibit a locally compact abelian group G which decomposes as (3.1.4) abstractly for closed subgroups H and N, but not topologically. Example 3.1.6 achieves this by choosing H and N to be discrete, whereas G is not. Example 3.1.6. We take G to be the direct product between a copy of the compact additive group Zp of p-adic integers for some odd prime number p, and a discrete copy Γ of the self-same group Zp (in other words, Γ is Zp as an abstract group, but with discrete topology). Now, in G = Zp × Γ we have a diagonal subgroup

H = {(g, g) | g ∈ Zp} Chapter86 3. Isomorphism theorems, modular law: the locally compact case

as well as an anti-diagonal one,

N = {(g, −g) | g ∈ Zp}.

We have H ∧ N = {0} because Zp is torsion-free, and also H + N = G because Zp is divisible by 2. Moreover, H and N are easily seen to both be closed in G and discrete. By construction, though, G is not. The preceding discussion explains why this will do.

3.2 The third isomorphism theorem

Recall ([34, §3.3, Theorem 19]) that this states that given normal subgroups N and H of G with N ≤ H, we have H/N E G/N and moreover (G/N)/(H/N) =∼ G/H.

Consider now the typical setup for the third isomorphism theorem: a locally compact quantum group G, and normal closed quantum subgroups N ≤ H of G. Then, because the composition N → H → G → G/N is trivial, Lemma 1.3.21 ensures that we have a factorization

G H G/N H/N of the top composition H → G → G/N. We will now examine bottom right morphism H/N → G/N. In general, we say that a morphism Π: P → Q of locally compact quantum groups has trivial kernel if the quotient quantum group

P → P/kerΠ of [53, Deﬁnition 4.4] is an isomorphism. Let us recal that Π: P → Q induces a morphism Π1 : P/ ker Π → Q which has trivial kernel. Lemma 3.2.1. The canonical morphism H/N → G/N has trivial kernel. Proof. Let us consider the morphism

η : H → G → G/N.

Using Proposition 3.1.1 we see that ker η = H ∧ N = N. In particular the kernel of the induced morphism H/ ker η → G/N is trivial.

Lemma 3.2.2. The closed image of the canonical map G/N → G/H is full. Proof. As noted in Example 1.3.14 the closed image of Π: G → G/H is full. Since N ⊂ H = ker Π the induced morphism G/N → G/H exists and its closed image coincides with the one of Π: G → G/H thus it is also full. Let us gather up all of the ingredients we have so far in the form of Lemma 3.2.1 and Lemma 3.2.2 into a weak version of the third isomorphism theorem (to be improved on later): 3.2. The third isomorphism theorem 87

Proposition 3.2.3. Given normal closed quantum subgroups N ≤ H of a locally compact quantum group G, the canonical morphism Π1 : H/N → G/N has trivial kernel and its closed image is precisely the kernel of Π2 : G/N → G/H.

Proof. Let Π: Gb → Hb be the morphism which is dual to the embedding H ≤ G. Let us ∞ ∞ recall that L (G/N) being a Baaj-Vaes subalgebra of L (G) can be interpreted as G[/N ≤ Gb . Using Lemma 1.3.15 to Π: Gb → Hb and G[/N ≤ Gb we conclude that the right quantum ∞ ∞ ¯ ∞ group homomorphism αH→G/N :L (G/N) → L (G/N) ⊗ L (H) is the restriction of the ∞ ∞ ¯ ∞ ∞ right quantum group homomorphism αH→G :L (G) → L (G) ⊗ L (H) to L (G/N) ⊂ ∞ L (G). Using Lemma 3.2.1 we conclude that the kernel of the morphism H → G/N is ∞ ∞ ¯ ∞ equal N and using Lemma 1.3.10 we get αH→G/N(L (G/N)) ⊂ L (G/N) ⊗ L (H/N). ∞ ∞ ¯ ∞ ∞ Summarizing the restriction of αH→G :L (G) → L (G) ⊗ L (H) to L (G/N) induces right quantum group homomorphism

∞ ∞ ¯ ∞ αH/N→G/N :L (G/N) → L (G/N) ⊗ L (H/N).

In particular ∞ ∞ 1 L (G/H) = {x ∈ L (G/N): αH/N→G/N(x) = x ⊗ }. (3.2.1)

Recalling that H/N → G/N is denoted by Π1 let us consider imΠ1 ≤ G/N. Equation (3.2.1) then shows that

∞ ∞ ∞ L ((G/N)/imΠ1) = cd(L (im[Π1)) = L (G/H). (3.2.2)

/ On the other hand noting that Π2 : G/N → G/H is represented by the bicharacter WG H ∈ ∞ ∞ ∞ ∞ L (G[/H) ⊗¯ L (G/N) (where we used that L (G/H) ⊂ L (G/N)) we get

∞ G/H ∞ σ−cls ∞ L ((G/N)/ ker Π2) = {(ω ⊗ id)(W ): ω ∈ L (G[/H)∗} = L (G/H) which together with (3.2.2) shows that ker Π2 = imΠ1.

In order to have a full analogue of [34, §3.3, Theorem 19], we would further want to know that the canonical morphism H/N → G/N identiﬁes the former with a closed quantum subgroup of the latter. Moreover, in view of Proposition 3.2.3 and Theorem 1.3.11, this amounts to showing that the action of H/N on G/N is integrable. To this end, we will ﬁrst need the following Weyl-integral-formula-type result.

Proposition 3.2.4. Given a normal closed quantum subgroup N E G a left-invariant Haar weight ϕG can be expressed as

ϕG/N ◦ T, where ∞ ∞ id⊗ϕ L (G) ⊗ L (N) N ∞ ∞ (3.2.3) L (G) L (G/N) T is a faithful semiﬁnite normal operator-valued weight. Chapter88 3. Isomorphism theorems, modular law: the locally compact case

Proof. The fact that the composition (3.2.3) is a faithful normal operator-valued weight (in the sense of [93, Definition IX.4.12]) into the right hand side (one needs to check that ∞ it lands in the algebra of N-invariants of L (G)) is essentially [97, Proposition 1.3]. The integrability [53, Theorem 6.2] of the action of the closed subgroup N on G means by definition that T is semifinite, and hence pre-composing with T turns semifinite weights ∞ ∞ on L (G/N) into semifinite weights on L (G) (see also e.g. [64, Definition 8.1]). Finally, the requisite invariance property of ϕG/N ◦T is a routine computation, using the invariance properties of ϕG/N and T .

Given a morphism Π: P → Q of locally compact quantum groups, we will denote by TP→Q the operator-valued weight

id⊗ϕ ∞ ∞ ∞ P ∞ L (Q) L (Q) ⊗¯ L (P) L (Q).

Let us note that in general TP→Q is not semiﬁnite. Finally, Proposition 3.2.4 will help in proving the missing integrability ingredient we remarked on above:

Proposition 3.2.5. Given closed normal subgroups N ≤ H of a locally compact quantum group G, the canonical action of H/N on G/N is integrable.

Proof. We have to show that the operator-valued weight TH/N→G/N deﬁned as

id ⊗¯ ϕ ∞ ∞ ∞ H/N ∞ L (G/N) L (G/N) ⊗¯ L (H/N) L (G/N)

∞ is semifinite, or equivalently, that there is at least one element of L (G/N) that is integrable with respect to the (H/N)-action ([64, Proposition 6.2]). We have already observed via [53, Theorem 6.2] that actions of closed quantum subgroups are integrable, and hence TN→G is semifinite. Similarly, TH→G is semifinite. We ∞ + will argue that for any x ∈ L (G) that is H-integrable, its image

∞ TN→G(x) ∈ L (G/N) is (H/N)-integrable; as observed, this is sufficient to finish the proof of the proposition. First, consider the following diagram of operator-valued weights and von Neumann algebra homomorphisms, where commutativity is immediate from the definitions:

L∞( ) ⊗¯ L∞( ) ⊗¯ L∞( ) G H N id⊗ϕN

∞ ∞ ∞ ∞ L (G) ⊗¯ L (H) L (G) ⊗¯ L (H/N)

∞ L (G) (3.2.4)

∞ ∞ ∞ ∞ L (G) ⊗¯ L (N) L (G/N) ⊗¯ L (H/N) id⊗ϕN ∞ L (G/N) 3.2. The third isomorphism theorem 89

Now further glue the commutative square

id⊗ϕ ∞ ∞ H/N ∞ L (G) ⊗ L (H/N) L (G) (3.2.5)

id⊗ϕ ∞ ∞ H/N ∞ L (G/N) ⊗ L (H/N) L (G/N) to the right hand side of (3.2.4). Using the Weyl integration formula (Proposition 3.2.4) for the normal subgroup N E H, we can see that the composition of the top half of (3.2.4) with the top horizontal arrow of

(3.2.5) yields precisely the semiﬁnite operator-valued weight TH→G. The commutativity of the compound diagram obtained by gluing (??) then proves our assertion that the image ∞ through TN→G of an H-integrable element of L (G) is (H/N)-integrable, thus completing the proof.

In summary, we obtain

Theorem 3.2.6. Let N ≤ H E G be inclusions of closed locally compact quantum subgroups, and assume furthermore that N is normal in G. Then, we have ∼ H/N E G/N and (G/N)/(H/N) = G/H. Proof. As noted above, Theorem 1.3.11 and Proposition 3.2.3 reduce the problem to showing that the action of H/N on G/N resulting from the canonical morphism H/N → G/N is integrable. This is exactly what Proposition 3.2.5 says.

In fact, some of the above results generalize somewhat so as to allow us to recover standard results on topological groups in the locally compact quantum setting. For instance, the conclusion that H/N → G/N is a closed embedding does not actually require the normality of H, and hence Theo- rem 3.2.6 extends to this general setup.

Theorem 3.2.7. Let N ≤ H ≤ G be closed embeddings of locally compact quantum groups, with N normal in G. Then the canonical morphism H/N → G/N is a closed embedding, and

∞ ∞ L ((G/N)/(H/N)) = L (G/H) Let us now brieﬂy go back to the setup of Proposition 3.1.1: H and N are closed quantum subgroups of G, with N normal. Then, by Lemma 1.3.21, the composition H → G → G/N always factors as

G H G/N H/H ∧ N

Moreover, Theorem 3.2.7 ensures that we can regard H ∨ N/N as a closed subgroup of G/N, and an examination of the proof of Proposition 3.1.1 shows that we actually have the following ampliﬁcation of Proposition 3.1.1. Chapter90 3. Isomorphism theorems, modular law: the locally compact case

Theorem 3.2.8. Let H ≤ G and N E G be closed quantum subgroups of a locally compact quantum group. Then, the canonical morphism

H/H ∧ N → G/N has trivial kernel and closed image H ∨ N/N ≤ G/N.

3.3 The modular law and Zassenhauss lemma

We now proceed to address an analogue of the Zassenhauss lemma for locally compact quantum groups. First, recall the classical (non-topological) statement, for instance as in [62, Vol. 1, p. 77].

0 0 Proposition 3.3.1. Let A E A and B E B be subgroups of a group G. Then, we have a canonical isomorphism A0(A ∩ B) B0(A ∩ B) =∼ . A0(A ∩ B0) B0(A0 ∩ B) Remark 3.3.2. The statement implicitly includes the claims that the group products appearing in the formula (such as A0(A ∩ B)) are indeed subgroups of G, and the groups appearing as denominators are normal in those appearing as numerators.

Recall ([62, Vol. 1, p. 77]) that the proof typically proceeds through the second isomorphism theorem for groups (which Proposition 3.1.1 replicates) by using it to implement connecting isomorphisms

A0(A∩B) B0(A∩B) A0(A∩B0) B0(A0∩B) A∩B =∼ (A0∩B)(A∩B0) =∼

We will adopt a similar approach here, but we need some preparatory remarks. First, note that it is implicit in the proof sketch we have just recalled that under the assumptions of the Zassenhaus lemma we have e.g.

(A ∩ B) ∩ (B0(A0 ∩ B)) = (A ∩ B0)(A0 ∩ B).

Given that A0 ∩ B is a normal subgroup of A ∩ B and normalizes B0, this follows from the modularity law for subgroups which we will use in the following form:

L ≤ H ≤ G, M ≤ G and L normalizes M ⇒ H ∩ ML = (H ∩ M)L.

Theorem 3.3.4 is an analogue of modularity in the locally compact quantum setting. Let us ﬁrst prove an easy inclusion.

Lemma 3.3.3. Let G be a locally compact quantum groups M, H ≤ G, and L ≤ H. Then

(H ∧ M) ∨ L ≤ H ∧ (M ∨ L).

Proof. Let us note that H∧M ≤ H∧(M∨L). Moreover, by assumption L ≤ H and clearly L ≤ M∨L thus L ≤ H∧(M∨L). This altogether shows that (H∧M)∨L ≤ H∧(M∨L). We now turn to suﬃcient conditions for an inclusion reversal in Lemma 3.3.3. The material surrounding Deﬁnition 1.3.31 above will be needed here. 3.3. The modular law and Zassenhauss lemma 91

Theorem 3.3.4. Let L ≤ H and M be closed quantum subgroups of a locally compact quantum group G such that L normalizes M. Then, we have

H ∧ (M ∨ L) = (H ∧ M) ∨ L. (3.3.1) if either

(a) L is compact, or

(b) H is open in G. Proof. We address the two versions of the result separately.

Proof of part (a). Here we rephrase the desired conclusion in terms of the quantum homogeneous spaces A• for • = H, M, etc (see notation in §1.3.1). Since cd is an anti- isomorphism of lattices the sought-after conclusion is

AH ∨ (AM ∧ AL) = (AH ∨ AM) ∧ AL. (3.3.2) Using lemma 3.3.3 we see that the right hand side contains the left hand side. We hence focus on proving the opposite inclusion. Let us ﬁrst consider the case when H and M are relatively well positioned in the sense of Deﬁnition 1.3.31. Now, as in the proof of Proposition 3.2.5, consider the operator-valued weights ∞ T•→G :L (G) → A•.

Since L is assumed to be compact, T = TL→G actually restricts to the identity on AL, and hence also on the right hand side of (3.3.2). On the other hand, in order to study the result of applying T to the right hand side algebra of (3.3.2), it suﬃces by (1.3.25) to look at products

x = xHxM, xH ∈ AH, xM ∈ AM When applied to the latter, due to the preservation by T of bimodule structures over

AH ⊆ AL, T produces the element xHT (xM).

We would be ﬁnished if we could show that T (xM) ∈ AM ∧ AL; this is what Lemma 3.3.6 below does. In order to drop the well positioning assumption let us consider D = H ∧ (L ∨ M). The ∞ ∞ ∞ ∞ ∞ ∞ containment L (Hb ) ∧ (L (Mc) ∨ L (Lb)) ⊂ (L (Hb ) ∧ L (Mc)) ∨ L (Lb), which is eﬀectively proved above under the well positioning assumption of H and L, is equivalent with the following containment

∞ ∞ ∞ ∞ ∞ ∞ L (Db) ∧ (L (Mc) ∨ L (Lb)) ⊂ (L (Db) ∧ L (Mc)) ∨ L (Lb). (3.3.3)

Since D, L, M ≤ L∨M, proving (3.3.3) we can substitute M∨L for G. After this substitution the normalization assumption of M by L gets replaced by the normality of M in G. Using Proposition 1.3.32 we see that D and M are well positioned and by the ﬁrst part of the proof (3.3.3) holds, thus we are done.

Proof of part (b). Here, we translate the claim into an equivalent statement for the underlying von Neumann algebras of the dual groups Gb , Hb , etc. Chapter92 3. Isomorphism theorems, modular law: the locally compact case

Since L normalizes M we can use Lemma 1.3.18 and get

∞ ∞ ∞ σ−cls L (M\∨ L) = {L (Mc)L (Lb)} .

∞ In conclusion, the von Neumann subalgebra of L (Gb ) corresponding to the left hand side of (3.3.1) is ∞ ∞ ∞ σ−cls L (Hb ) ∧ {L (Mc)L (Lb)} . Similarly, the right hand side corresponds to

∞ ∞ ∞ σ−cls {(L (Hb ) ∧ L (Mc)) L (Lb)} , and we seek to prove

∞ ∞ ∞ σ−cls ∞ ∞ ∞ σ−cls L (Hb ) ∧ {L (Mc)L (Lb)} = {(L (Hb ) ∧ L (Mc)) L (Lb)} . (3.3.4)

As in the ﬁrst part, the inclusion of the right hand side in the left hand side is Lemma 3.3.3, and we only need to prove ‘⊆’. We will use essentially the same strategy as in the proof of part (1), substituting for ∞ TL→G :L (G) → AL from that other proof the expectation

∞ ∞ T :L (Gb ) → L (Hb ) corresponding to the compatible Haar weights on the two von Neumann algebras (this is where the openness of H is essential; see e.g. [51, Theorem 7.5]). As before, applying T to the left hand side of (3.3.4) on the one hand acts as the identity, and on the other produces from a product

∞ ∞ x = xMxL, xM ∈ L (Mc), xL ∈ L (Lb) the element

T (xM)xL ∞ due to the L (Lb)-bimodule map property of T . The conclusion that x = T (x) belongs to the right hand side of (3.3.4) now follows from the fact that

∞ ∞ T (xM) ∈ L (Hb ) ∧ L (Mc), which in turn relies on Proposition 1.3.26.

Remark 3.3.5. The reader should note that since L = H ∧ L, the modular law is really a form of the distributive law H ∧ (M ∨ L) = (H ∧ M) ∨ (H ∧ L). The latter, however, is false in general.

Lemma 3.3.6. Let G be a locally compact quantum group, L ≤ G a compact quantum subgroup, and M ≤ G a closed quantum subgroup normalized by L. Then, the expectation

∞ T :L (G) → AL leaves AM invariant. 3.3. The modular law and Zassenhauss lemma 93

Proof. Indeed, the normalization condition ensures that the right action of L on G descends to an action on the quantum homogeneous space G/M via the commutative diagram (see Lemma 1.3.19) ∞ ∞ ∞ L (G) L (G) ⊗¯ L (L)

¯ ∞ AM AM ⊗ L (L) The conclusion now follows from the deﬁnition of the expectation T as the coaction

∞ ∞ ∞ L (G) → L (G) ⊗¯ L (L) followed by an application of the Haar state ψL to the right hand tensorand. Remark 3.3.7. We note that an appropriately rephrased version of Lemma 3.3.6 holds under the weaker requirement that L/L∧M acts integrably on G/M. T would then restrict to a semiﬁnite operator-valued weight

AM → AL ∧ AM. We do, however, need compactness in the proof of Theorem 3.3.4 above, where the operator- valued weight T had to be an expectation and hence ﬁx AL pointwise. Note that Theorem 3.3.4 does not hold in full generality, even for classical locally compact abelian groups. In order to see this, we can modify Example 3.1.4 as follows.

4 Example 3.3.8. Our ambient group G = T × R is written as in Example 3.1.4, multi- plicatively in the ﬁrst four variables and additively in the last. We then take ixθ ixθ M = {(e 1 , ··· , e 4 , x) | x ∈ R} and isφ L = {(e , 1, 1, 1, s) | s ∈ R}. with φ and θi linearly independent over Q. Finally, let

isφ itφ H = {(e , e , 1, 1, s + t) | s, t ∈ R}.

H is easily seen to be a two-dimensional closed Lie subgroup of G that contains L and intersects M trivially. Since ML is dense in G, we have H ∧ (M ∨ L) = H on the left hand side of (3.3.1). On the other hand, the right hand side (H ∧ M) ∨ L is L. As in Chapter2 above, we have the following consequence of modularity (i.e. of Theorem 3.3.4).

0 0 Proposition 3.3.9. Let A E A and B E B be either (a) compact or

(b) open quantum subgroups of a locally compact quantum group G. Then, we have an isomorphism 0 ∨ ( ∧ ) 0 ∨ ( ∧ ) A A B =∼ B A B . A0 ∨ (A ∧ B0) B0 ∨ (A0 ∧ B) Chapter94 3. Isomorphism theorems, modular law: the locally compact case

Proof. This follows from Theorem 3.3.4 in much the same way in which Proposition 2.2.11 follows from Proposition 2.2.4 and Proposition 2.2.10, by applying the earlier result to 0 0 H = A ∧ B, L = A ∧ B and M = A . Everything goes through as before, modulo the observation that in the open case we need Proposition 1.3.26 in order to conclude that A ∧ B is open in the open subgroup B, and hence is also open in G by Proposition 1.3.28.

0 Remark 3.3.10. In case (a) of Proposition 3.3.9 it is enough that L (and hence say B ) be compact.

3.4 Schreier and Jordan-Hölder-type results

We devote this section to certain partial analogues of Theorems 2.3.3, 2.4.3 and 2.4.4 in the setting of locally compact quantum groups. In this context, the relevant notions of (sub)normal series and reﬁnements thereof make sense virtually verbatim, so we point to Deﬁnitions ?? for a reminder. We write {G`}`≥0 for the generic (sub)normal series

G = G0 ≥ G1 ≥ G2 ≥ G3 ≥ · · · ≥ Gk = 1. (3.4.1) of closed quantum subgroups of a locally compact quantum group G.

Theorem 3.4.1. Let G be a locally compact quantum group. Then, any two subnormal 0 series {G`} and Gt of G admit equivalent reﬁnements, provided

0 G`, ` ≥ 1 and Gt, t ≥ 1 are

(a) compact or

(b) open.

Proof. One can simply imitate the proof of Theorem 2.3.3, making use of parts (a) and (b) of Proposition 3.3.9 respectively for the two parts of the present result.

As for an analogue of Theorem 2.4.3 and Proposition 2.4.4, we have

Theorem 3.4.2. Let G be a locally compact quantum group. Then, all composition series of G consisting of (a) compact or

(b) open quantum subgroups are equivalent.

Proof. Just as the proof of Theorem 2.4.3, this follows mechanically once we have Theo- rem 3.4.1 above.

The compact versions of Theorems 3.4.1 and 3.4.2 refer to subnormal series (3.4.1) in which all G`, ` ≥ 1 are compact, but G = G0 need not be so. Let us note that this is equivalent to the subquotient G/G1 being non-compact. Indeed, we have 3.4. Schreier and Jordan-Hölder-type results 95

Proposition 3.4.3. A locally compact quantum group G is compact if and only if it admits a subnormal series (3.4.1) with compact quotients Gi/Gi+1. Proof. The direct implication ‘⇒’ is immediate by considering the trivial length-zero series, so we focus on the opposite implication. By induction, it suﬃces to show that if L E G is compact along with G/L, then so is G. This in turn follows from the fact that by the Weyl integration formula proven above (Proposition 3.2.4) the Haar weight of G is a state.

Chapter 4

Classiﬁcation results for the compact quantum group O−1(2)

In this chapter, we will give ﬁrst the list of ergodic actions of O−1(2) and we will apply the results of Chapter1 Section 1.4 to obtain the list of embeddable ergodic actions and the list of idempotent states of O−1(2).

4.1 Ergodic actions of O−1(2)

In this section, we will give the complete list of ergodic coactions of O−1(2). Let’s recall [6, Theorem 4.3]:

−1 Theorem 4.1.1. The category of corepresentations of C(On ) is tensor equivalent to the category of representations of On.

By Theorem 4.1.1, the compact quantum groups O−1(2) and O(2) are monoidally equivalent and by Theorem 1.4.19, the ergodic actions of O(2) are in bijective correspondence with the ergodic actions of O−1(2). Then it is suﬃcient to classify the ergodic α actions of O(2). Recall that if G is a compact group, H a closed subgroup, and H y N an ergodic action of H on a von Neumann algebra N, we have the induced ergodic action of G on

G ∞ IndH (N) = {f ∈ L (G, N) | ∀g ∈ G, h ∈ H, αh(f(gh)) = f(g)} ⊆ L∞(G)⊗¯ N, given by 0 −1 0 (αg(f))(g ) = f(g g ).

∗ αh(x) = π(h)xπ(h) .

For example, when π is the trivial representation (or a one-dimensional representation), G ∞ we have IndH (C) = L (G/H) with the left translation action. ∼ In the following, we will identify O(2) = Z2 n T, with T = {z ∈ C | |z| = 1} the circle group, and with the cyclic group Z2 = {1, σ} acting on T by σ(z) =z ¯. 98 Chapter 4. Classification results for O−1(2)

∼ O(2) Theorem 4.1.2. Let O(2) M be an ergodic action. Then M = Ind (B(Vπ)) for yα H some closed subgroup H ⊆ O(2) and π an irreducible representation of H.

Proof. For n ∈ Z, we write Mn for the n-th spectral subspace of M with respect to the n action of T, i.e. αz(x) = z x for x ∈ Mn. In particular, M0 is the space of T-invariant elements. As M0 is stabilised by σ, and as Z2 must act ergodically on M0 since O(2) acts ∼ ∼ ergodically on M, we must have either M0 = C with σ acting trivially or M0 = C ⊕ C with σ interchanging the factors. In the first case, T already acts ergodically on M0. If then for some n ∈ Z we have ∗ ∗ Mn 6= {0}, it follows that for a non-zero x ∈ Mn we have x x ∈ M0, and hence x x = λ ∈ ∗ 0 0 C\{0}. Similarly, xx = λ ∈ C\{0}. We must hence have λ = λ and x a scalar multiple of ∗ a unitary. If y is another non-zero element in Mn, we must have also x y a scalar multiple of the unit, hence y ∈ Cx. It follows that all spectral subspaces are one-dimensional. Moreover, if u ∈ Mn and v ∈ Mm are non-zero, then uv ∈ Mm+n non-zero. Hence either Mn is zero for all n 6= 0, and M = C, or there exists a least n ∈ N \{0} such that Mn 6= 0. k In the latter case, any non-zero spectral subspace is of the form Mkn = Cu for k ∈ Z and u a fixed unitary element in Mn. In particular, it follows that M is commutative, and hence of the form L∞(O(2)/H) for H a closed subgroup of O(2). In the second case, let p be a non-trivial projection in M0. Then also the central support z(p) of p in M must be a projection in M0. Hence there are two cases: z(p) = p or z(p) = 1. In the first case, p is central in M, and hence M = pM ⊕ (1 − p)M with the action of T on both pM and (1 − p)M ergodic. By the first part of the proof, it follows that then M is commutative, and hence of the form L∞(O(2)/H) for H a closed subgroup of O(2). In the second situation, it follows that MpM is weakly dense in M. We must have then as well that M(1 − p)M is weakly dense in M, and hence we can decompose M as ! AB M = , CD where A = pMp, B = pM(1 − p), C = (1 − p)Mp and D = (1 − p)M(1 − p). In particular, A, D are von Neumann algebras with ergodic actions of T, and hence commutative. Writing ασ = σ, then as σ(p) = 1 − p, it follows that σ(A) = D and σ(B) = C, and we can write ! AB M = . σ(B) σ(A)

! 0 x It follows also that B = {0}, as otherwise would be a non-zero, non-scalar 0 σ(x) 0 O(2)-invariant element in M for any non-zero x ∈ B0, contradicting ergodicity. On the other hand, not all Bn can be zero, as B is non-zero (since p is not central). Now as before, we can show that each non-zero Bn must be one-dimensional, and spanned by an element ∗ ∗ u such that u u = 1 − p and uu = p. Fixing a non-zero Bl and such a u, we further have that B = uu∗B ⊆ uσ(A) ⊆ B, and hence B = uσ(A). Similarly, B = Au. If now A = A0 = Cp, it follows that also σ(A) = σ(A)0 = C(1−p), and B and σ(B) are one-dimensional, i.e. M = M2(C). But an ergodic action O(2) y M2(C) determines an 2 2 O(2) 2 irreducible projective representation of O(2) on C , hence M = B(C ) = IndO(2)(B(C )). In case where A is not one-dimensional, we must have that A =∼ L∞(SO(2)/H) for some ﬁnite cyclic group H = Zk ⊆ SO(2). As A then contains non-zero spectral subspaces For a projection p ∈ M the smallest projection q in the center Z(M) of M such that p ≤ q is called the central support and is denoted by z(p). 4.1. Ergodic actions of O−1(2) 99

precisely at the values kZ, it follows that B contains non-zero spectral subspaces at the values in l + kZ. In particular, as B0 is zero, we must have l not a multiple of k, and we can assume 0 < l < k. Take v a unitary in Ak. It follows that the direct sum M of all O(2)-spectral subspaces ! vr 0 has a basis consisting of partial isometries of either the form v(r) = , v(r) = + 0 0 − ! ! ! 0 0 0 uvr 0 0 , u(r) = and u(r) = , where σ(v(r)) = v(r) and 0 σ(v)r + 0 0 − σ(u)σ(v)r 0 ± ∓ (r) (r) σ(u± ) = u∓ , and where T acts by

(r) ±kr (r) (r) ±(l+rk) (r) αz(v± ) = z v± , αz(u± ) = z u± . ! ! a b a zlb But let = , acting ergodically on B( 2) by α = and Dk Z2 n Zk C z c d z−lc d ! ! a b d c O(2) 2 ασ = . Then it is easy to see that Ind (B( )) is spanned by the partial c d b a Dk C isometries ! ! z−kr 0 0 0 V (r)(z) = ,V (r)(σz) = , + 0 0 + 0 z−kr

(r) (r) V− (g) = V+ (σg) and ! ! 0 z−l−kr 0 0 U (r)(z) = ,U (r)(σz) = , + 0 0 + z−l−kr 0

U (r)(g) = U (r)(σg). We then have a unique O(2)-equivariant ∗-isomorphism IndO(2)(B( 2)) → − + Dk C (r) (r) (r) (r) M such that V± 7→ v± and U± 7→ u± .

It remains to determine isomorphisms between the induced actions. (∞) The irreducible (projective) representations of O(2) give either the trivial action β0 (∞) on C (for the characters) or the ergodic action βl/2 = α on M2(C) by ! ! ! ! a b a zlb a b d c α = , α = z c d z−lc d σ c d b a for l ∈ N \{0} (with the even l coming from representations and the odd l coming from non-trivial projective representations). (∞) 2 The irreducible representations of T all give the same action α = α on C = L∞(O(2)/T ), namely

αz((x, y)) = (x, y), ασ(x, y) = (y, x).

(k) The irreducible representations of Zk for some ﬁxed k all give the same action α = α ∞ ∞ ∞ on L (O(2)/Zk) = L (T/Zk) ⊕ L (T/Zk), namely

αz(f, g) = (fz, gz), ασ(f, g) = (g, f), where fz denotes the z-translate of f. 100 Chapter 4. Classification results for O−1(2)

(k) Finally, for the Dk we have the action α = β0 coming from the characters of Dk, ∞ giving the action on L (O(2)/Dk), or for the non-character (projective) representations (k) (k) the actions βl/2 where 0 < l < k a natural number and βl/2 the induced action coming from the (projective) Dk-representation ! ! ! ! a b a zlb a b d c α = , α = . z c d z−lc d σ c d b a

Hence, we obtain:

Proposition 4.1.3. The full list of non-equivalent ergodic actions of O(2) is:

(k) (∞) (∞) (k0) 0 0 {βl/2 , βl0/2 , α , α | k, k ∈ N0, l ∈ N, 0 ≤ l < k}. Question 4.1.4. Does monoidal equivalence allow us to determine the ergodic action αb α Xσ x O−1(2) corresponding to each family of ergodic action X x O(2)? ( ∼ G × M → M Consider an action α : G → Aut(M) = αb : of a group G on a von (g, m) 7→ αg(m) ( M → M ⊗ L∞(G) =∼ L∞(G, M) Neumann algebra M. This induces a map αb : such m 7→ (g 7→ αg(m)) that: αb(m)(g) ∈ M. In fact, it easy to verify that αb is an homomorphism and veriﬁes (αb ⊗ id) ◦ αb = (id ⊗ ∆) ◦ αb. That is the map αb is a coaction on a compact quantum group.

4.2 Embeddable ergodic actions of O−1(2)

In this section we determine the embeddable ergodic actions on O−1(2), based on those of O(2) classified above in Proposition 4.1.3. The plan for achieving this is as follows. First, note that by definition an embeddable ergodic action is by definition a comodule ∗-algebra of the CQG algebra A−1 associated to O−1(2) which embeds into A−1 as such (i.e. by an embedding that preserves all of the structure: comodule, algebra, etc.). Since the twisting equivalence λ. (see the notation from Section 4.4) that implements Theorem 1.4.19 also implements an equivalence between the categories of coideal ∗-algebras over A−1 and the untwisted version A (algebra of representative functions on the classical group O(2)), it will be sufficient to identify the ergodic O(2)-action B in the list of Proposition 4.1.3 for which ∼ λ . B = A−1 as A−1 comodule ∗-algebras, and to then also identify the members of that list that embed into B. (2) We will see that there is only one candidate for B (namely β1/2) using the Peter- Weyl theorem to determine the representation type of the ergodic actions identified in Proposition 4.1.3 (where by representation type we mean the multiplicities of the various irreducible O(2)-representations). Indeed, this is the substance of the following result.

Proposition 4.2.1. The only comodule algebras among those in Proposition 4.1.3 that (1) ∼ (2) are isomorphic to A as O(2)-representations are α = A itself and β1/2. Proof. The (∞)-superscript O(2)-representations are ﬁnite-dimensional, so we can dis- count them for the purposes of this proposition. 4.2. Embeddable ergodic actions of O−1(2) 101

For the other members of the list, we will use the Frobenius reciprocity formula

O(2) ∼ homO(2)(V, IndH W ) = homH (V,W ) (4.2.1) for V ∈ RepO(2) and W ∈ RepH in order to compute the multiplicities of various irreducible O(2)-representations. For each k ≥ 1 we have a 2-dimensional O(2)-representation Vk whose restriction to T, upon identifying the Pontryagin dual

∼ Tb = Z, splits as k ⊕ (−k). (k) Now, for k ≥ 2, α is induced from the non-trivial cyclic group Zk ⊂ T. Taking H = Zk, W to be trivial, and V = V1 in (4.2.1), the right hand side vanishes and hence (k) so must the left hand side. This means that V1 is not a summand of α , k ≥ 2, and hence these list members can also be dropped as candidates for an isomorphism to A as A-comodules. (k) Next we look at the representations β0 for all k ≥ 1 induced from the trivial representation of the order-2k dihedral groups Dk ⊂ O(2). In these cases, (4.2.1) with H = Dk, W trivial and V being the non-trivial character of O(2) annihilates the right hand side, and hence the left hand side too. In conclusion, the non-trivial character of O(2) does not (k) appear in β0 ; this disqualiﬁes these representations. (k) Finally, we consider β`/2 for ` > 0 and k ≥ 2. Here, we apply (4.2.1) with H = Dk, W the representation of Dk on M2 described in the discussion preceding Proposition 4.1.3, and V = V1. There are now a few possibilities: (a) If ` > 1 then the right hand side of (4.2.1) is zero, so these cases can be discarded; (b) If ` = 1 and k ≥ 3 then the right hand side of (4.2.1) is one-dimensional, because (k) the restriction of V1 to Dk is irreducible. In conclusion V1 appears in β`/2 with multiplicity one, but it appears in A with multiplicity two(by Peter-Weyl, since it is a two-dimensional irreducible representation). Once more, these cases do not qualify for the purposes of the proposition; (c) Finally, ` = 1 and k = 2 is left, in which case one easily checks that the multiplicities 2 match as expected. Indeed, Dk is then the Klein group (Z/2) , and its 4-dimensional (2) representation W that is induced up to O(2) to produce β1/2 breaks up as a sum of all of its characters. It follows from the previous paragraph that if the irreducible O(2)-representation V is one-dimensional then the right hand side of (4.2.1) is also one-dimensional, whereas if V is two-dimensional then its restriction to D2 breaks up as a sum of two distinct characters, and hence the right hand side of (4.2.1) is two-dimensional. This ﬁnishes the proof of the proposition.

We can now record the consequence alluded to above.

Corollary 4.2.2. The twisting equivalence λ. induces a bijection between

(k) (k) {α , βl/2 | k = ∞ or even , l = 0 or odd } from Proposition 4.1.3 and the isomorphism classes of embeddable ergodic actions of O−1(2). 102 Chapter 4. Classification results for O−1(2)

Proof. The function algebra of O−1(2) can be obtained from that of O(2) by twisting the multiplication both on the right and the left, by the cocycle λ and its convolution inverse λ−1. Since λ. by deﬁnition twists by λ on the right, the A-comodule algebra B from the introductory remarks to Section 4.2 is a twist of A on the left and hence cannot be abelian, and yet must have the same representation type as A as a right A-comodule. It must thus (2) be β1/2 by Proposition 4.2.1. In summary, the desired conclusion will follow once we show that the ergodic O(2)- (2) actions listed in the statement are precisely those that embed into β1/2. Throughout the proof, we denote by W the D2-representation on M2 that gives rise (2) to β1/2 by induction to O(2). We examine the representations listed in Proposition 4.1.3 systematically.

Type-α actions. (∞) (2) α is two-dimensional and clearly embeds into β1/2, as the diagonal subalgebra of the realization of W as 2 × 2 matrices. As for α(k) for positive integers k, consider ﬁrst the case when k is odd. If we had an embedding (k) (2) α ⊆ β1/2, then the Frobenius adjunction (4.2.1) would turn it into a map

O(2) (k) ResH α → W (4.2.2)

of algebras in RepD2 . The condition that k be odd then ensures that this map is surjective, (k) since in that case all four characters of D2 appear in the restriction of α . Since however the left hand side of (4.2.2) is commutative while the right hand side is not, we obtain a contradiction. (k) (2) For even k on the other hand, we can embed α into β1/2 by inducing in stages. First, embed IndDk ⊆ IndDk W Zk C D2 as the diagonal subalgebra of the 2×2 matrix realization of W , and then induce the entire embedding further to O(2).

Type-β actions, l = 0. (∞) (2) β0 is simply the trivial representation and hence is embeddable into β1/2. We note (k) (k) also that β0 for odd k can be eliminated in exactly the same way we did α above. (k) (2) (k) For even k β0 is again embeddable into β1/2 by the case of even α , since we have

(k) (k) β0 ⊂ α .

Type-β actions, l > 0. (k) Consider the case of βl/2 (including k = ∞) for even positive l. Here we have an embedding (∞) (k) βl/2 ⊆ βl/2 (4.2.3) (2) of algebras in RepO(2), and hence an embedding of the right hand side into β1/2 would imply the existence of a morphism of the left hand side into W in the category RepD2 . 4.3. Idempotent states of O−1(2) 103

This is impossible: both the left hand side of (4.2.3) and W are 2 × 2 matrix algebras and hence the morphism would have to be one-to-one, but the evenness of l ensures that when restricted to D2 the left hand side of (4.2.3) has a two-dimensional space of invariants. When k is positive and odd, then for every l we have an even l0 such that l0 ≡ l (modk). We have an embedding (∞) (k) βl0/2 ⊆ βl/2 (k) of algebras in RepO(2) and we can repeat the argument above to conclude that βl/2 is not (2) embeddable into β1/2. For even k (including by abuse the case k = ∞ with Dk = O(2)) and positive odd l (∞) the restriction of βl/2 to Dk embeds into

IndDk W, D2 (k) (2) and hence βl/2 is embeddable into β1/2, as desired. This concludes the last case and the proof of the result.

4.3 Idempotent states of O−1(2)

In this section, we will deduce from Section 4.2, the list of idempotent states of O−1(2). In fact, by Example 1.4.3 and Lemma 1.4.14, there exists then an order-preserving bijection between the embeddable ergodic action of O−1(2) and the idempotent states on O−1(2). Having succeeded to make the list of the embeddable ergodic actions of O−1(2) and since by definition, an embeddable ergodic action is identified with a coideal subalgebra and since according to the fundamental result [39, Theorem 4.1] by Franz-Skalski, there is a bijective correspondence between the expected coideal subalgebras and the idempotent states of a compact quantum group; Our project has met a final concern: How to express by a concrete formula an idempotent state from an action of a compact quantum group In fact, although we characterized them by listing embeddable ergodic action using [39], we still have to list explicitly the idempotent states of O−1(2) from the list of its embeddable ergodic actions. Considering that: 1. Let us recall that if one takes the comultiplication itself as the action (i.e the quantum group itself is taken as the coideal subalgebra), it is not immediate to produce the corresponding Haar state from the multiplication.

2. Also if we consider the subalgebra C, then we see that we can not produce the Haar state from the action which is simply C → C ⊗ C, 1 7→ 1 ⊗ 1. The problem is that starting from the proof of [39, Theorem 4.1], there are several elements that we have abstractly but non-explicitly, and they depend on each other. Indeed, to obtain the idempotent state of the coideal subalgebra, one needs the conditional expectation E; And to get E, we need the state. We believe that if there is a chance to obtain a concrete formula of the idempotent state from an embeddable ergodic action, it will be by expressing the conditional expectation E as a function of the embeddable ergodic action α only by not involving the state itself, then compose E with the counit ε. This question summarizes our point of view. 104 Chapter 4. Classification results for O−1(2)

Question 4.3.1. Is it possible to obtain such a concrete formula in general for a compact quantum group at least in the Kac case?

4.4 A counterexample

In the section, we will show that in general, one can not extend the principal result of [30] for embeddable ergodic actions, answering Question 1.4.1 (or rather to the more precise version of it we give below) in the negative. Speciﬁcally, let G1 and G2 be two monoidally equivalent compact quantum groups. Suppose furthermore that the monoidal equivalence is implemented by twisting by a cocycle ⊗2 λ : C(G1) → C. Then, according to the proof of Theorem 1.4.19, we have a twisting operation λ. that implements an equivalence between the category Erg(G1) of ergodic G1-actions and the analogous category Erg(G2). In this setting, the formulation of Question 1.4.1 that we answer here is Question 4.4.1. Does λ. : Erg(G1) → Erg(G2) restrict to an equivalence between subcategories of embeddable ergodic coactions?

To describe the counterexample (and explain why it is one), we start out by taking G1 to be the order-64 Heisenberg group

4 4 4 H4 := hε1, ε2, δ | ε1 = ε2 = δ = 1, δε1 = ε1δ, ε2δ = δε2, ε1ε2 = δε2ε1i.

Its deformation G2 (with underlying function algebra H = C(H4)) will be the one used in [16], described in some detail in Section 6 of that paper. Although we will not write the cocycle ⊗ λ : C(H4) → C explicitly, recall that it is obtained as the composition

λ ⊗2 C(H4) C C(Γ)⊗2 where the left hand arrow is restriction along the inclusion

2 ∼ 2 2 (Z/2) = Γ := hε1, ε2i ⊂ H4 and the right hand arrow is some cocycle inducing the non-zero cohomology class in

2 × ∼ H (Γb, C ) = Z/2. We refer to [16, Section 6] for a deduction of the needed structural results on H, which we will now recollect. It will be more convenient to dualize and work with the group algebra ∗ CH4 and its deformed version H , which coincides with CH4 as an algebra but has twisted comultiplication. First, the order-16 subgroup

2 ∼ 2 2 (Z/2) × (Z/4) = G := hε1i × hε2i × hδi 4.4. A counterexample 105

of H4 survives the deformation, in the sense that the new comultiplication on CH4 pre- ∗ serves the inclusion CG ⊂ CH4, and hence the left hand side is a Hopf subalgebra of H . The elements of G are precisely the grouplikes of H∗. The balance of 64−16 = 48 dimensions of H∗ consists of twelve 2×2 matrix coalgebras, corresponding to twelve simple 2-dimensional comodules. Setting η = δ2, these coalgebras are the images of ±1 ±1 Di = span{εi , εi η}, i = 1, 2 and −1 −1 −1 −1 D3 = span{ε1ε2, ε1 ε2 , ε1ε2η, ε1 ε2 η} through the action of G by left multiplication; each Dj, j = 1, 2, 3 has a size-4 orbit under this action. Now, using the correspondence between coideals of a quantum group and those of its dual (e.g. [55]), right coideal ∗-subalgebras of C(H4) and H (which embeddable ergodic coactions map isomorphically onto) correspond in an order-reversing fashion to the coideal ∗ ∗-subalgebras of CH4 and H . Moreover, if the coideal ∗-subalgebra A of H corresponds to cd(A) ∈ H∗, then we have

dim(A) · dim(cd(A)) = dim(H) = 64.

This follows for instance from the fact that every coideal subalgebra A ⊂ H we are considering is a ﬁnite-dimensional C∗-algebra and hence Frobenius, which implies [70] that H is free as a right A-module, say H =∼ A⊕m. But then the dual

∗ ∼ ⊕m cd(A) = H/Hker(ε|A) = C

dim(H) as vector spaces, giving the desired conclusion (since m = dim(A) ). All in all, if the answer to Question 4.4.1 were aﬃrmative, then we would have a dimension and inclusion-preserving bijection between the coideal ∗-subalgebras of CH4 and those of H∗ (note that the preceding discussion was needed in order to pass from coideal subalgebras of C(H4) and H to those of their duals). Moreover, because G ⊂ H4 is untwisted, this bijection would preserve G along with its subgroups. This is precisely what we will show is not the case. Speciﬁcally, we have

Proposition 4.4.2. The only 4-dimensional coideal ∗-subalgebra of H∗ that does not ∗ 2 2 2 intersect CG ⊂ H and which contain Chε1i is the group algebra of hε1i × hε2i. Proof. Let A ⊂ H∗ be a four-dimensional coideal subalgebra as in the statement, diﬀerent from 2 Chεi i. (4.4.1) We have already noted that G accounts for all of the grouplikes of H∗; since the group (4.4.1) is the only order-4 subgroup of G that does not intersect the center Z(G) = Z(H4) = hδi, it follows that as a right H∗-comodule, A must break up as

2 C ⊕ Cε1 ⊕ V for some two-dimensional irreducible comodule V . Since moreover A is a subalgebra, V 2 must be invariant under tensoring with Cε1. The description of the coalgebra structure of H∗ given above then implies that we have

2 2 V ⊂ D1, δD1, ε2D1, or δε2D1. 106 Chapter 4. Classification results for O−1(2)

Now, it is shown in [16, Lemma 6.2] that

( 1 1 a := 2 ε1 + ε1η b := 2 ε1 − ε1η 1 −1 −1 1 −1 −1 c := 2 ε1 − ε1 η d := 2 ε1 + ε1 η

2 are matrix counits for D1, and ε1V = V then implies that we have

V = span{a + c, b + d},V = span{a − c, d − b},

2 2 or their images under multiplication by δ, ε2 or δε2. But it is easily checked that in 2 ∗ none of these cases is C ⊕ Cε1 ⊕ V a subalgebra of H . For instance, when V ⊂ D1, the 2 2 decomposition of (a + c) in one case and (a − c) in the other with respect to the H4 basis 2 contains ε1η; analogous arguments work in the other three cases. Consequently,

Corollary 4.4.3. There does not exist an inclusion, dimension, and G-preserving bijection ∗ between the coideal subalgebras of CH4 and those of H .

Proof. This follows from Proposition 4.4.2 and the fact that CH4 has an additional coideal subalgebra as in the statement of that result, namely the group algebra of the subgroup ∼ hε1i = Z/4 of H4. Note however that the answer to Question 4.4.1 is affirmative for those compact quantum groups G = G1 whose only ergodic comodule ∗-algebra admitting all irreducible comodules of O(G) with full multiplicity is O(G) itself. The latter comodule algebras (usually referred to as full multiplicity ergodic actions) 2 1 are classified by the cohomology set H (Gb , S ) of the discrete quantum dual of G described for instance in [77, §2.4]. This is an extension to the quantum realm of the classification carried out in [108] in terms of cocycles.

2 1 Remark 4.4.4. Note that when G = Γb for an ordinary discrete group Γ, the set H (Gb , S ) 2 1 is simply the standard 2-cohomology group H (Γ, S ) of Γ valued in the trivial Γ-module 1 S .

As one ﬁnal simple observation, we remark that if G1 happens to have trivial 2- 2 1 cohomology set H (Gc1, S ), the answer to Question 4.4.1 is aﬃrmative:

Proposition 4.4.5. Let G1 and G2 be two monoidally equivalent compact quantum groups, 2 1 and assume H (Gc2, S ) is trivial. Then,

λ. : Erg(G1) → Erg(G2) restricts to an equivalence between subcategories of embeddable ergodic actions.

Proof. The functor λ. implements a bijection between the isomorphism classes of the ergodic coideal algebras of O(G1) embeddable into O(G1) as coideal algebras and the ergodic coideal ∗-algebras of G2 embeddable into λ.O(G1). The latter is a full-multiplicity ergodic action of G2, and hence by assumption it must be (isomorphic to) O(G2) itself. Bibliography

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